Similarly to the definition of module [31], a hyper module can be defined as
follows.

###### Definition VII.1

Let $R$ be a hyper ring. A (left) $R$-module is an additive (abelian) hyper
group $A$ with a function $\pi:R\times A\rightarrow A$ such that

  * (i)

$\displaystyle r(a+b)=ra+rb,\quad r\in R,\;a,~{}b\in A;$ (194)

  * (ii)

$\displaystyle(r+s)a=ra+sa,\quad r,s\in R,\;a,~{}b\in A;$ (195)

  * (iii)

$\displaystyle r(sa)=(rs)a.$ (196)

If $R$ has an identity set ${\bf e}_{R}$ for the product, such that

$\displaystyle e_{R}x\sim x\quad e_{R}\in{\bf e}_{R},\forall x\in A,$ (197)

and there exists at least one $e_{x}\in{\bf e}_{R}$, such that

$\displaystyle e_{x}x=x,$ (198)

then $A$ is said to be a unitary hyper R-module.

###### Remark VII.2

  * (i)

A module can be considered as a particular hyper module, where the $R$ is
degenerated to a ring.

  * (ii)

The equality (197) can equivalently be written as

$\displaystyle e_{R}x-x\in{\bf e},\quad e_{R}\in{\bf e}_{R}.$ (199)

  * (iii)

We are particularly interested in unitary (hyper-)R-module.

Consider the (unitary) module of (hyper-)matrix ring with (hyper-)vector
group.

####  VII-A1 (Hyper) Vector Group

  * (i)

$\displaystyle{\mathcal{A}}^{V}_{1}:=({\mathbb{R}}^{n},+),\quad
n\in{\mathbb{Z}}_{+},$ (200)

where $+$ is the classical vector addition. ${\mathcal{A}}^{V}_{1}$ is a
vector group with unique identity $0_{n}$.

  * (ii)

$\displaystyle{\mathcal{A}}^{V}_{2}:=({\mathbb{R}}^{\infty},\hat{+}),$ (201)

where $\hat{+}$ is defined as follows: Let $x\in{\mathbb{R}}^{m}$,
$y\in{\mathbb{R}}^{n}$, and $t=\operatorname{lcm}(m,n)$. Then

$\displaystyle x\hat{\pm}y:=\left(x\otimes{\bf
1}_{t/m}\right)+\left(y\otimes{\bf 1}_{t/n}\right)\in{\mathbb{R}}^{t}.$ (202)

Then ${\mathcal{A}}^{V}_{2}$ is a hyper-vector group with identity set

${\bf e}=\\{e_{n}=0_{n}\;|\;n\in{\mathbb{Z}}_{+}\\}.$

  * (iii)

$\displaystyle{\mathcal{A}}^{V}_{3}:=({\mathbb{R}}^{\infty},\bar{+}),$ (203)

where $\bar{+}$ is the defined as follows: Let $x\in{\mathbb{R}}^{m}$,
$y\in{\mathbb{R}}^{n}$, and $t=\operatorname{lcm}(m,n)$. Then

$\displaystyle
x\hat{\pm}y:=\left(x\otimes{\mathcal{E}}_{t/m}\right)+\left(y\otimes{\mathcal{E}}_{t/n}\right)\in{\mathbb{R}}^{t}.$
(204)

Then ${\mathcal{A}}^{V}_{3}$ is a hyper-vector group with identity set

${\bf e}=\\{e_{n}=0_{n}\;|\;n\in{\mathbb{Z}}_{+}\\}.$

####  VII-A2 (Hyper) Matrix Ring

  * (i)

$\displaystyle{\mathcal{R}}^{M}_{1}:=({\mathcal{M}}^{n\times
n},+,\times),\quad n\in{\mathbb{Z}}_{+},$ (205)

where $+$ and $\times$ are the classical matrix addition and production
respectively. ${\mathcal{R}}^{M}_{1}$ is a matrix ring with unique identity
$I_{n}$.

  * (ii)

$\displaystyle{\mathcal{R}}^{M}_{2}:=(\overline{\mathcal{M}}^{m\times
n},+,\,\rotatebox{-90.0}{$\ltimes$}\,),\quad m,n\in{\mathbb{Z}}_{+},$ (206)

where $+$ is the classical matrix addition and the  $\ltimes$ is the DK-STP,
defined by (40). ${\mathcal{R}}^{M}_{2}$ is a matrix ring with the unique
identity $I_{m\times n}$.

  * (iii)

$\displaystyle{\mathcal{R}}^{M}_{3}:=({\mathcal{M}}_{1},\bar{+},\ltimes),$
(207)

where $\bar{+}$ is defined by (29) and the $\ltimes$ is defined by (18).
${\mathcal{R}}^{M}_{3}$ is a matrix unitary hype-ring with the identity set

${\bf e}=\\{0_{n\times n}\;|\;n\in{\mathbb{Z}}_{+}\\};$

and

${\bf e}_{R}=\\{I_{n}\;|\;n\in{\mathbb{Z}}_{+}\\}.$

  * (iv)

$\displaystyle{\mathcal{R}}^{M}_{4}:=({\mathcal{M}}_{1},\hat{+},\circ),$ (208)

where $\hat{+}$ is defined by (35) and the $\circ$ is defined by (19).
${\mathcal{R}}^{M}_{4}$ is a matrix unitary hype-ring with the identity set

${\bf e}=\\{0_{n\times n}\;|\;n\in{\mathbb{Z}}_{+}\\};$

and

${\bf e}_{R}=\\{I_{n}\;|\;n\in{\mathbb{Z}}_{+}\\}.$

  * (v)

$\displaystyle{\mathcal{R}}^{M}_{5}:=(\overline{\mathcal{M}}_{\mu},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}),$
(209)

where the $\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$ is defined by (51).
${\mathcal{R}}^{M}_{5}$ is a matrix unitary hype-ring with the identity set

${\bf e}=\\{0_{n\mu_{y}\times n\mu_{x}}\;|\;n\in{\mathbb{Z}}_{+}\\};$

and

${\bf e}_{R}=\\{I_{n}\;|\;n\in{\mathbb{Z}}_{+}\\}.$

  * (vi)

$\displaystyle{\mathcal{R}}^{M}_{6}:=(\overline{\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}).$
(210)

${\mathcal{R}}^{M}_{6}$ is a matrix unitary hype-ring with the identity set

${\bf e}=\\{0_{m\times n}\;|\;m,n\in{\mathbb{Z}}_{+}\\};$

and

${\bf e}_{R}=\\{I_{m\times n}\;|\;m,n\in{\mathbb{Z}}_{+}\\}.$

####  VII-A3 (Hyper) Matrix Module

We construct the following combinations of matrix (hyper) rings with vector
(hyper) groups.

  * (i)

$\aleph_{1}:=({\mathcal{R}}^{M}_{1},{\mathcal{A}}^{V}_{1},\times)$, where
$\times$ is the classical matrix-vector product.

  * (ii)

$\aleph_{2}:=({\mathcal{R}}^{M}_{2},{\mathcal{A}}^{V}_{2},\,\rotatebox{-90.0}{$\ltimes$}\,)$
(or
$({\mathcal{R}}^{M}_{2},{\mathcal{A}}^{V}_{3},\,\rotatebox{-90.0}{$\ltimes$}\,)$
), where  $\ltimes$ is the DK-STP, defined by (40).

  * (iii)

$\aleph_{3}:=({\mathcal{R}}^{M}_{3},{\mathcal{A}}^{V}_{2},\ltimes)$ (or
$({\mathcal{R}}^{M}_{3},{\mathcal{A}}^{V}_{3},\ltimes)$).

  * (iv)

$\aleph_{4}:=({\mathcal{R}}^{M}_{4},{\mathcal{A}}^{V}_{2},\circ)$ (or
$({\mathcal{R}}^{M}_{4},{\mathcal{A}}^{V}_{3},\circ)$ .

  * (v)

$\aleph_{5}:=({\mathcal{R}}^{M}_{5},{\mathcal{A}}^{V}_{2},\,\rotatebox{-90.0}{$\ltimes$}\,)$
(or
$({\mathcal{R}}^{M}_{5},{\mathcal{A}}^{V}_{3},\,\rotatebox{-90.0}{$\ltimes$}\,)$).

  * (vi)

$\aleph_{6}:=({\mathcal{R}}^{M}_{6},{\mathcal{A}}^{V}_{2},\,\rotatebox{-90.0}{$\ltimes$}\,)$
(or$({\mathcal{R}}^{M}_{6},{\mathcal{A}}^{V}_{3},\,\rotatebox{-90.0}{$\ltimes$}\,)$).

###### Proposition VII.3

  * (i)

The $\aleph_{1}$ and $\aleph_{2}$ are unitary modules.

  * (ii)

The $\aleph_{3}$, $\aleph_{4}$, $\aleph_{5}$ and $\aleph_{6}$ are unitary
hyper-modules.

Proof. We prove $\aleph_{2}$, the proof of others is similar. What do we need
to verify is (194)-(196). In fact, (195) is trivial, and (196) is well know
[19]. Hence, we prove (194) only.

Let $A\in{\mathcal{M}}_{m\times n}$, $x\in{\mathbb{R}}^{p}$,
$y\in{\mathbb{R}}^{q}$, $p\vee n=\alpha$, $q\vee n=\beta$, $p\vee q=t$, and
$n\vee p\vee q=s$. Then

$\begin{array}[]{l}A\,\rotatebox{-90.0}{$\ltimes$}\,x=\left(A\otimes{\bf
1}^{T}_{\alpha/n}\right)\left(x\otimes{\bf
1}_{\alpha/p}\right)\in{\mathbb{R}}^{m};\\\
A\,\rotatebox{-90.0}{$\ltimes$}\,y=\left(A\otimes{\bf
1}^{T}_{\beta/n}\right)\left(y\otimes{\bf
1}_{\beta/q}\right)\in{\mathbb{R}}^{m}.\\\ \end{array}$
$\begin{array}[]{l}A\,\rotatebox{-90.0}{$\ltimes$}\,(x\hat{+}y)=A\,\rotatebox{-90.0}{$\ltimes$}\,\left(x\otimes{\bf
1}_{t/p}+y\otimes{\bf 1}_{t/q}\right)\\\ ~{}=\left(A\otimes{\bf
1}^{T}_{s/n}\right)\left(x\otimes{\bf 1}_{s/p}+y\otimes{\bf 1}_{s/q}\right)\\\
~{}=\left(A\otimes{\bf 1}^{T}_{s/n}\right)\left(x\otimes{\bf
1}_{s/p}\right)+\left(A\otimes{\bf 1}^{T}_{s/n}\right)\left(y\otimes{\bf
1}_{s/q}\right)\\\ ~{}=\left(A\otimes{\bf
1}^{T}_{\alpha/n}\right)\left(x\otimes{\bf
1}_{\alpha/p}\right)+\left(A\otimes{\bf
1}^{T}_{\beta/n}\right)\left(y\otimes{\bf 1}_{\beta/q}\right)\\\ \end{array}$

(194) follows.

$\Box$

###  VII-B Linear (Control) System

###### Definition VII.4

Let $\aleph=({\mathcal{R}}_{0},{\mathcal{A}}_{0},\pi)$ be a (hyper-) matrix-
vector (MV-) module.

  * (i)

A discrete-time linear system is defined as

$\displaystyle\begin{array}[]{l}x(t+1)=M(t)x(t),\quad
x(0)=x_{0}\in{\mathcal{V}}_{0},\\\ \quad
M(t)\in{\mathcal{R}}_{0},\;x(t)\in{\mathcal{A}}_{0},\end{array}$ (213)

where $M(t)x(t)=\pi(M(t),x(t))$.

  * (ii)

A continuous-time linear system is defined as

$\displaystyle\begin{array}[]{l}\dot{x}=M(t)x(t),\quad
x(0)=x_{0}\in{\mathcal{V}}_{0},\\\ \quad
M(t)\in{\mathcal{R}}_{0},\;x(t)\in{\mathcal{A}}_{0}.\end{array}$ (216)

  * (iii)

A discrete-time linear control system is defined as

$\displaystyle\begin{array}[]{l}x(t+1)=M(t)x(t)+B(t)u(t),\quad
x(0)=x_{0}\in{\mathcal{V}}_{0},\\\ \quad
M(t),B(t)\in{\mathcal{R}}_{0},\;x(t),u(t)\in{\mathcal{A}}_{0}.\end{array}$
(219)

  * (iv)

A continuous-time linear control system is defined as

$\displaystyle\begin{array}[]{l}\dot{x}=M(t)x(t)+B(t)u(t),\quad
x(0)=x_{0}\in{\mathcal{V}}_{0},\\\ \quad
M(t),B(t)\in{\mathcal{R}}_{0},\;x(t)\in{\mathcal{A}}_{0}.\end{array}$ (222)

###### Remark VII.5

  * (i)

When $M(t)=M$ (and $B(t)=B$ for control systems) the systems are called the
time-invariant linear (control) systems.

  * (ii)

We are only interested in the case when
$\aleph\in\\{\aleph^{i}\;|\;i\in[1,6]\\}$, which were defined in the previous
subsection.

  * (iii)

When $\aleph=\aleph^{1}$, the systems are the classical linear (control)
systems, which have been investigated over 8 decades [48].

  * (iv)

When $\aleph=\aleph^{3}$ or $\aleph=\aleph^{4}$, the systems have been
investigated by [10]. Particularly, the trajectories of the time-invariant
systems have been constructed.

###### Example VII.6

Consider a linear control system

$\displaystyle\dot{x}=Ax(t)+Bu(t)+C\eta(t),$ (223)

where $x(t)\in{\mathbb{R}}^{n}$ is the state, $u(t)\in{\mathbb{R}}^{m}$ is the
control, $\eta(t)\in{\mathbb{R}}^{r}$ is the disturbance,
$A\in{\mathcal{M}}_{n\times n}$, $B\in{\mathcal{M}}_{n\times
m}$,$C\in{\mathcal{M}}_{n\times r}$. Assume $\eta(t)$ is a dimension-varying
disturbance, then we may replace (223) by

$\dot{x}=Ax(t)+Bu(t)+C\,\rotatebox{-90.0}{$\ltimes$}\,\eta(t).$

Then the system can handle disturbance with various dimensions. If there are
some state variable dimension-varying vibrations, say,

$x(t_{\tau})=x(t_{\tau_{-}}),\quad\tau\in\\{t_{1},t_{2},\cdots\\},$

Then (223) can be replaced by

$\dot{x}=A\,\rotatebox{-90.0}{$\ltimes$}\,x(t)+Bu(t)+C\,\rotatebox{-90.0}{$\ltimes$}\,\eta(t).$

##  VIII Hyper Vector Space

Up to now the algebraic structures of MVMDs have been investigated. From this
section on, we will consider the geometric structure of MVMDs.

###  VIII-A From Hyper Group to Hyper Vector Space

To be precise we first review the concept of vector space in linear algebra.

###### Definition VIII.1

[29] A vector space ${\mathcal{V}}$ over ${\mathbb{R}}$ is a triple
${\mathcal{V}}=(X,+,\cdot)$ , where $X$ is a set with elements $x\in X$ are
vectors; addition $+:X\times X\rightarrow X$, and scalar product
$\cdot:{\mathbb{R}}\times X\rightarrow X$, satisfying

  1. (1)

$(X,+)$ is an abelian group, that is,

     * (i)

Associativity:

$\displaystyle(x+y)+z=x+(y+z),\quad x,y,z\in X.$ (224)

     * (ii)

Commutativity:

$\displaystyle x+y=y+x,\quad x,y\in X.$ (225)

     * (iii)

Zero: There exists a unique $\vec{0}\in X$, such that

$\displaystyle x+\vec{0}=\vec{0}+x=x,\quad\forall x\in X.$ (226)

     * (iv)

Inverse: For each $x\in X$ there exists a $-x\in X$, such that

$\displaystyle x+(-x)=0,\quad x\in X.$ (227)

  2. (2)

Scalar product satisfies

     * (i)

Associativity:

$\displaystyle(r_{1}r_{2})\cdot x=r_{1}\cdot(r_{2}\cdot x),\quad
r_{1},r_{2}\in{\mathbb{F}},~{}x\in X.$ (228)

     * (ii)

Distributive Law:

$\displaystyle\begin{array}[]{l}(r_{1}+r_{2})\cdot x=r_{1}\cdot x+r_{2}\cdot
x,\quad r,r_{1},r_{2}\in{\mathbb{R}},\\\ r\cdot(x+y)=r\cdot x+r\cdot
y,\quad~{}x,~{}y\in X.\end{array}$ (231)

     * (iii)

Unit:

$\displaystyle 1\cdot x=x,\quad 1\in{\mathbb{R}},x\in X.$ (232)

When the MVMDs are considered, it is natural to consider “pseudo-” vector
space, which is defined by modifying the definition of vector space a little
bit.

###### Definition VIII.2

A hyper-vector space ${\mathcal{V}}$ over ${\mathbb{R}}$ is a triple
${\mathcal{V}}=(X,+,\cdot)$ as in Definition VIII.1, satisfying

  1. (1’)

$(X,+)$ ia an abelian hyper group.

  2. (2)

Same as the (2) in Definition VIII.1.

Recall that each hyper group has its component groups and as the operator (it
is addition now) is consistent w.r.t. equivalence then the equivalence group
exists. Correspondingly, we have the following result.

###### Proposition VIII.3

Consider a hyper-vector space ${\mathcal{V}}=(X,+,\cdot)$. Assume ${\bf
e}=\\{e_{\lambda}\;|\;\lambda\in\Lambda\\}$. Then

  * (i)

For each $\lambda\in\Lambda$, $X_{\lambda}$ is a vector space.

  * (ii)

Assume the addition is consistent w.r.t. equivalence, then the equivalence
group is a vector space.

Proof.

  * (i)

Let $x\in V_{\lambda}$. Then,

$xe_{\lambda}=e_{\lambda}x=x,\quad
x^{-1}e_{\lambda}=e_{\lambda}x^{-1}=x^{-1}.$

It follows that for any $\alpha\in{\mathbb{R}}$

$\begin{array}[]{l}(\alpha x)e_{\lambda}=e_{\lambda}(\alpha x)=\alpha x,\\\
(\alpha x)^{-1}e_{\lambda}=e_{\lambda}(\alpha x)^{-1}=(\alpha
x)^{-1}.\end{array}$

Hence $\alpha x\in X_{\lambda}$, which ensures that $V_{\lambda}$ is a vector
space.

  * (ii)

Since

$\overline{\alpha x}=\alpha\bar{x},\quad\alpha\in{\mathbb{R}},$

$(\bar{V},+)$ is also a vector space.

$\Box$

###  VIII-B Hyper Vector Space over Matrices

It was mentioned in Remark III.9 that all the addition hyper groups on MVMDs
are abelian. In addition, for MVMDs the scaler product is naturally defined.
Moreover, the properties of scalar product required for vector spaces are
automatically satisfied. Then based on Proposition III.8 the following results
are obvious.

###### Proposition VIII.4

  * (i)

$({\mathbb{R}}^{\infty},\hat{+})$ is a hyper-vector space.

  * (ii)

$({\mathcal{M}}_{1},\bar{+})$ is a hyper-vector space.
$({\mathcal{M}}_{1},\hat{+})$ is also a hyper-vector space.

  * (iii)

$({\mathcal{M}}_{\mu}.\hat{+})$ is a hyper-vector space with $\hat{+}$ defined
by (30), where the identity set is the same as (63).

  * (iv)

$({\mathcal{M}},\hat{+})$ is a hyper-vector space.

The decomposition for hyper groups can be immediately extended to hyper-vector
space. Then we have

###### Proposition VIII.5

Let $(X,+,{\bf e})$ be a hyper-vector space.

  * (i)

For each $e\in{\bf e}$, $(X_{e},+)$ is a vector space, called the component
subspace of the hyper-vector space.

  * (ii)

Assume the equivalence is consistent with operator $+$ and scalar product,
then the quotient space is also a vector space called the equivalence vector
space.

##  IX Metric Space Structure on ${\mathbb{R}}^{\infty}$

###  IX-A Hyper Inner Product Space

Replacing vector space by hyper-vector space, the hyper inner product space is
defined as follows [44].

###### Definition IX.1

A hyper-vector space $X$ over ${\mathbb{F}}$ (${\mathbb{F}}={\mathbb{C}}$ or
${\mathbb{F}}={\mathbb{R}}$) is called an hyper-inner product space if to each
pair of elements $x,~{}y\in X$ there is associated a number $\langle
x,y\rangle\in{\mathbb{F}}$, called the hyper-inner product of $x$ and $y$,
with the following properties:

  * (i)

$\displaystyle\langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle,\quad
x,y,z\in X.$ (233)

  * (ii)

$\displaystyle\langle x,y\rangle=\overline{\langle y,x\rangle}.$ (234)

  * (iii)

$\displaystyle\langle ax,y\rangle=a\langle x,y\rangle.$ (235)

  * (iv)

$\displaystyle\langle x,x\rangle\geq 0,~{}\mbox{and}~{}\langle x,x\rangle\neq
0,~{}\mbox{if}~{}x\not\in{\bf 0}.$ (236)

Consider ${\mathbb{R}}^{\infty}$, an inner product has been defined as
follows.

###### Definition IX.2

[10] Consider $x,y\in{\mathbb{R}}^{\infty}$, say, $x\in{\mathbb{R}}^{m}$,
$y\in{\mathbb{R}}^{n}$, and $\operatorname{lcm}(m,n)=t$ The inner product of
$x$ and $y$ is defined by

$\displaystyle\begin{array}[]{l}\langle
x,y\rangle_{{\mathcal{V}}}:=\frac{1}{t}x\vec{\cdot}y\\\
=\frac{1}{t}\left(x\otimes{\bf 1}_{t/m}\right)^{T}\left(y\otimes{\bf
1}_{t/n}\right).\end{array}$ (239)

Recall that $({\mathbb{R}}^{\infty},\hat{+},\cdot)$ is a hyper-vector space,
it is readily verifying that the inner product defined by (239) is a hyper-
inner product.

Note that by Definition IX.2 if $x\in{\mathbb{R}}^{n}$, then

$\displaystyle\langle
x,x\rangle_{{\mathcal{V}}}=\frac{1}{n}x^{T}x=\frac{1}{n}\langle x,x\rangle,$
(240)

where $\langle x,x\rangle$ is the Euclidean inner product. Hence the inner
product defined by (239) has the “normalized” Euclidean inner product. This is
very important for comparing the vectors in Euclidean spaces of various
dimensions.

Alternatively, we can also define

$\displaystyle\langle
x,y\rangle_{{\mathcal{V}}}:=\left(x\otimes{\mathcal{E}}_{t/m}\right)^{T}\left(y\otimes{\mathcal{E}}_{t/n}\right).$
(241)

For a hyper-inner product space, the inner-product is said to be consistent
with the operator (addition) if $x_{1}\sim x_{2}$ and $y_{1}\sim y_{2}$ imply

$\langle x_{1},y_{1}\rangle=\langle x_{2},y_{2}\rangle.$

The following result is an immediate consequence of the definition.

###### Proposition IX.3

Let ${\mathcal{H}}:=({\mathcal{V}},\langle\cdot,\cdot\rangle)$ be a hyper-
inner product space, where ${\mathcal{V}}$ is a hyper-vector space and
$\langle\cdot,\cdot\rangle$ is the inner product. If the inner-product is
consistent with the addition, then the equivalence space, denoted by
$\overline{\mathcal{H}}:=(\overline{\mathcal{V}},\langle\cdot,\cdot\rangle)$,
is an inner product space, with the inner product as

$\displaystyle\langle\bar{x},\bar{y}\rangle:=\langle x,y\rangle.$ (242)

###### Example IX.4

Consider the hyper-inner product space
${\mathcal{H}}:=({\mathbb{R}}^{\infty},\hat{+},\langle\cdot,\cdot\rangle)$,
where the inner product is defined by (239). It has been proved in [10] that
this inner product is consistent with the addition, then it is clear that the
equivalence space
$\overline{\mathcal{H}}:=(\overline{{\mathbb{R}}}^{\infty},\hat{+},\langle\cdot,\cdot\rangle)$
is an inner-product space. It has also been proved in [10] that this inner-
product space is not complete, hence it is not a Hilbert space.

###### Definition IX.5

  1. (1)

A norm on a hyper-vector space $X$ is a real function, whose value at $x\in
X$, denoted by $\|x\|$, with the properties:

     * (i)

$\displaystyle\|x+y\|\leq\|x\|+\|y\|,\quad x,y\in X.$ (243)

     * (ii)

$\displaystyle\|ax\|=|a|\|x\|.$ (244)

     * (iii)

$\displaystyle\|x\|\geq 0,~{}\mbox{and}~{}\|x\|\neq
0,~{}\mbox{if}~{}x\not\in{\bf 0}.$ (245)

  2. (2)

Assume ${\mathcal{V}}$ is a hyper-inner product space, i.e., it is a hyper-
vector space with an inner product $\langle\cdot,\cdot\rangle$, then a norm
can be defined by

$\displaystyle\|x\|:=\sqrt{\langle x,x\rangle}.$ (246)

  3. (3)

The norm is said to be consistent with the equivalence, if $x\sim y$ implies
$\|x\|=\|y\|$.

A hyper-vector space with a norm is called a normed hyper-vector space.

The following is an immediate consequence of definitions.

###### Proposition IX.6

Consider a normed hyper-vector space. If the norm is consistent with the
equivalence, its equivalence space is a normed vector space.

###### Example IX.7

Consider the hyper-inner product space
${\mathcal{H}}:=({\mathbb{R}}^{\infty},\hat{+},\langle\cdot,\cdot\rangle)$.

  * (i)

Its norm has been defined in [10] as

$\displaystyle\|x\|_{{\mathcal{V}}}:=\sqrt{\left<x,x\right>_{{\mathcal{V}}}}.$
(247)

  * (ii)

It has been proved in [10] that the norm defined by (246) is consistent with
the equivalence. Hence the equivalence space
$(\overline{{\mathbb{R}}}^{\infty},\hat{+},\langle\cdot,\cdot\rangle)$ is a
normed vector space.

Let $X$ be a normed hyper-vector space, and $x,y\in X$. A natural way to
define the distance between $x$ and $y$ is

$\displaystyle d(x,y):=\|x-y\|.$ (248)

As an example, consider ${\mathbb{R}}^{\infty}$. Let
$x,y\in{\mathbb{R}}^{\infty}$. The distance of $x$ and $y$ is defined by [10]

$\displaystyle d_{{\mathcal{V}}}(x,y):=\|x-y\|_{{\mathcal{V}}}.$ (249)

It is readily verifying that the distance defined in (248), or particularly
(249) for ${\mathbb{R}}^{\infty}$, it is a hyper-metric, which turns the
hyper-vector space into a hyper-metric space.

###### Proposition IX.8

  * (i)

The metric defined by (249) is a hyper-metric, which makes
$({\mathbb{R}}^{\infty},d)$ a hyper-metric space.

  * (ii)

The hyper-metric space is a special pseudo-matric space.

  * (iii)

The metric defined by (249) is consistent with equivalence. That is, assume
$x_{1}\sim x_{2}$ and $y_{1}\sim y_{2}$, then $d(x_{1},y_{1})=d(x_{2},y_{2})$.

  * (iv)

The equivalence space $\bar{{\mathbb{R}}}^{\infty}$ is a metric space.

The topology deduced by the hyper-metric $d_{{\mathcal{V}}}$, defined by (249)
and denoted by ${\mathcal{T}}_{d}$, makes ${\mathbb{R}}^{\infty}$ a
topological space. This topological space has the following properties [10]:

###### Proposition IX.9

Consider the topological space $({\mathbb{R}}^{\infty},{\mathcal{T}}_{d})$.

  * (i)

It is separable. That is, there exists a countable subset, which is dense in
${\mathbb{R}}^{\infty}$.

  * (ii)

It is not a Hausdorff space. (In fact, it is not even $T_{0}$).

Finally, we briefly review its equivalence space. It is easy to verify the
following equivalent statements.

###### Proposition IX.10

Let $x,~{}y\in{\mathbb{R}}^{\infty}$. $x$ and $y$ are equivalent, if one of
the following equivalent statements holds.

  * (i)

$x,~{}y$ are equivalent, denoted by $x\leftrightarrow y$.

  * (ii)

There exist $0_{p}$ and $0_{q}$ such that $x\hat{+}0_{p}=y\hat{+}0_{q}$.

  * (iii)

There exist ${\bf 1}_{p}$ and ${\bf 1}_{q}$ such that $x\otimes{\bf
1}_{p}=y\otimes{\bf 1}_{q}$.

  * (iv)

$d_{{\mathcal{V}}}(x,y)=0$.

Denote the quotient space by

$\displaystyle\Omega:={\mathbb{R}}^{\infty}/\leftrightarrow.$ (250)

According to the Proposition IX.10, it is clear that $\Omega$ is both the
equivalence space and the topological quotient space. Moreover, this space is
a Hausdorff space.

We refer to [24] for related topics about topological spaces; and to [44, 25]
for some related knowledge for inner product space.

###  IX-B Geometric Structure on ${\mathbb{R}}^{\infty}$

First, we consider the projection from ${\mathbb{R}}^{m}$ to
${\mathbb{R}}^{n}$.

###### Definition IX.11

[10] Let $\xi\in{\mathbb{R}}^{m}$. The projection of $\xi$ on
${\mathbb{R}}^{n}$, denoted by $\pi^{m}_{n}(\xi)$, is defined as

$\displaystyle\pi^{m}_{n}(\xi):=argmin_{x\in{\mathbb{R}}_{n}}\|\xi\hat{-}x\|_{{\mathcal{V}}}.$
(251)

The projection can be expressed by matrix product form as follows.

###### Proposition IX.12

[10]

$\displaystyle
x_{0}:=\pi^{m}_{n}(\xi)=\Pi^{m}_{n}\xi\in{\mathbb{R}}^{n},\quad\xi\in{\mathbb{R}}^{m},$
(252)

where ($t=\operatorname{lcm}(m,n)$)

$\displaystyle\Pi^{m}_{n}=\frac{1}{t/n}\left(I_{n}\otimes{\bf
1}^{T}_{t/n}\right)\left(I_{m}\otimes{\bf 1}_{t/m}\right),$ (253)

Moreover, $x_{0}\bot\left(\xi\hat{-}x_{0}\right)$ (see Figure 6).

${\mathbb{R}}^{m}\ni\xi$$x_{0}$$x_{0}\hat{-}\xi\in{\mathbb{R}}^{t}$${\mathbb{R}}^{n}$
Figure 6: Cross-dimensional Projection

We give an example:

###### Example IX.13

Assume $m=4$, $n=6$. Let
$\xi=(\xi_{1},\xi_{2},\xi_{3},\xi_{4})^{T}\in{\mathbb{R}}^{m}$. We consider
its projection on ${\mathbb{R}}^{6}$. First, $t=\operatorname{lcm}(m,n)=12$,
then

$\begin{array}[]{l}\Pi^{4}_{6}=\frac{1}{2}\left(I_{6}\otimes{\bf
1}^{T}_{2}\right)\left(I_{4}\otimes{\bf 1}_{3}\right)\\\
=\begin{bmatrix}1&0&0&0\\\ 0.5&0.5&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0.5&0.5\\\
0&0&0&1\\\ \end{bmatrix}.\end{array}$

Hence,

$x_{0}=(\xi_{1},0.5\xi_{1}+0.5\xi_{2},\xi_{2},\xi_{3},0.5\xi_{3}+0.5\xi_{4},\xi_{4})^{T}.$
$\begin{array}[]{l}x_{0}\hat{-}\xi=(0,0,0.5\xi_{2}-0.5\xi_{1},0.5\xi_{1}-0.5\xi_{2},0,0\\\
~{}0,0,0.5\xi_{4}-0.5\xi_{3},0.5\xi_{3}-0.5\xi_{4},0,0)^{T}.\end{array}$

It is ready to check that

$\langle x_{0}\hat{-}\xi,x_{0}\rangle_{{\mathcal{V}}}=0.$

###### Theorem IX.14

(Generalized Pythagoras’ Theorem)

  * (i)

Let $\xi\in{\mathbb{R}}^{m}$, $x_{0}=\pi^{m}_{n}(\xi)\in{\mathbb{R}}^{n}$,
$x_{0}\hat{-}\xi\in{\mathbb{R}}^{t}$, where $t=\operatorname{lcm}(m,n)$. Then

$\displaystyle\|\xi\|_{{\mathcal{V}}}^{2}=\|x_{0}\|_{{\mathcal{V}}}^{2}+\|\xi-
x_{0}\|_{{\mathcal{V}}}^{2}.$ (254)

Figure 6 depicts the relationship of above vectors.

  * (ii)

Let $\xi\in{\mathbb{R}}^{m}$, and the projection of $\xi$ on
${\mathbb{R}}^{n}$ be $x_{0}=\pi^{m}_{n}(\xi)$. Then $\xi-x_{0}$ is
perpendicular to ${\mathbb{R}}^{n}$. That is, for any $z\in{\mathbb{R}}^{n}$
we have

$\displaystyle\langle(\xi\hat{-}x_{0}),(z-x_{0})\rangle_{{\mathcal{V}}}=0.$
(255)

Moreover,

$\displaystyle\|\xi\hat{-}z\|^{2}_{{\mathcal{V}}}=\|\xi\hat{-}x_{0}\|^{2}_{{\mathcal{V}}}+\|z\hat{-}x_{0}\|^{2}_{{\mathcal{V}}}.$
(256)

Figure 7 depicts the relationship of the vectors.

Proof. We prove (ii) only. The proof of (i) is similar.

Embed ${\mathbb{R}}^{n}$ into ${\mathbb{R}}^{t}$ by $x\mapsto x\otimes{\bf
1}_{t/n}$. Using Proposition IX.3, we have

$(\xi\hat{-}x_{0})\perp({\mathbb{R}}^{n}\otimes{\bf 1}_{t/n}).$

Then we have

$(\xi\hat{-}x_{0})\perp(z-x_{0}\otimes{\bf 1}_{t/n}).$

Using Pythagoras’ Theorem on ${\mathbb{R}}^{t}$, we have

$\|\xi\hat{-}z\|^{2}_{{\mathcal{V}}}=\|\xi\hat{-}x_{0}\|^{2}_{{\mathcal{V}}}+\|(z-x_{0})\otimes{\bf
1}_{t/n}\|^{2}_{{\mathcal{V}}}.$

Then Proposition IX.3 yields (256) immediately. $\Box$

$\xi\hat{-}z$$x_{0}$$\xi\hat{-}x_{0}\in{\mathbb{R}}^{t}$${\mathbb{R}}^{n}$$z$$z-x_{0}$$\xi\in{\mathbb{R}}^{m}$
Figure 7: Projection is orthogonal to ${\mathbb{R}}^{n}$

###### Example IX.15

Recall Example IX.13, a straightforward computation shows that

$\|\xi\|_{{\mathcal{V}}}^{2}=\frac{1}{4}\left(\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2}+\xi_{4}^{2}\right);$
$\begin{array}[]{l}\|x_{0}\|_{{\mathcal{V}}}^{2}=\frac{1}{6}\left(1.25\xi_{1}^{2}+1.25\xi_{2}^{2}+1.25\xi_{3}^{2}+1.25\xi_{4}^{2}\right.\\\
~{}\left.+0.5\xi_{1}\xi_{2}+0.5\xi_{3}\xi_{4}\right).\end{array}$
$\begin{array}[]{l}\|x_{0}\hat{-}\xi\|_{{\mathcal{V}}}^{2}=\frac{1}{12}\left(0.5\xi_{1}^{2}+0.5\xi_{2}^{2}+0.5\xi_{3}^{2}\right.\\\
~{}\left.+0.5\xi_{4}^{2}+\xi_{1}\xi_{2}+\xi_{3}\xi_{4}\right).\end{array}$

Using these equalities, (254) follows immediately.

Using projection and Pythagoras’ Theorem, more geometric structure over
${\mathbb{R}}^{\infty}$ can be explored.

###### Example IX.16

  * (i)

The relationship between ${\mathbb{R}}^{m}$ and ${\mathbb{R}}^{n}$ ($m\neq
n$)?

Assume $s=\gcd(m,n)$ is the great common divisor. Then it is easy to see that
under the topology ${\mathcal{T}}_{d}$, some points of ${\mathbb{R}}^{m}$ and
${\mathbb{R}}^{n}$ are glued together, when they are of zero distance.
Moreover,

$d(x,y)=0,\quad x\in{\mathbb{R}}^{m},y\in{\mathbb{R}}^{n},$

if and only if, there exists $z\in{\mathbb{R}}^{s}$ such that $x=z\otimes{\bf
1}_{m/s}$ and $y=z\otimes{\bf 1}_{n/s}$. Roughly speaking, ${\mathbb{R}}^{m}$
intersects ${\mathbb{R}}^{n}$ at a subspace $Z\sim{\mathbb{R}}^{s}$. Figure 8
shows this.

$Z$${\mathbb{R}}^{n}$${\mathbb{R}}^{m}$ Figure 8: Intersection of
${\mathbb{R}}^{m}$ and ${\mathbb{R}}^{n}$

  * (ii)

The sphere in ${\mathbb{R}}^{\infty}$:

Let $c_{0}\in{\mathbb{R}}^{\infty}$ and $r>0$. The sphere in
${\mathbb{R}}^{\infty}$, centering at $c_{0}$ with radius $r$, is defined by

$\displaystyle
S_{r}(c_{0}):=\left\\{x\in{\mathbb{R}}^{\infty}\;|\;d_{{\mathcal{V}}}(x,c_{0})=r\right\\}.$
(257)

Assume $c_{0}\in{\mathbb{R}}^{m}$. Now consider ${\mathbb{R}}^{n}$. We project
$c_{0}$ to ${\mathbb{R}}^{n}$, say, $\pi^{m}_{n}(c_{0})=c_{n}$. Denote
$h_{n}:=c_{0}-c_{n}$. Assume

$\displaystyle\|h_{n}\|_{{\mathcal{V}}}\leq r.$ (258)

Then a sphere can be constructed as

$\displaystyle
S_{n}:=\left\\{x\in{\mathbb{R}}^{n}\;|\;d(x,c_{n})=\sqrt{r^{2}-\|h_{n}\|^{2}_{{\mathcal{V}}}}\right\\}.$
(259)

We conclude that the sphere $S_{r}(c_{0})$ in ${\mathbb{R}}^{\infty}$ consists
of all spheres $S_{n}\in{\mathbb{R}}^{n}$, where the distance of
${\mathbb{R}}^{n}$ with $c_{0}$ satisfies (258) and $S_{n}$ is constructed by
(259). Figure 9 shows this.

${\mathbb{R}}^{n}$${\mathbb{R}}^{m}$${\mathbb{R}}^{r}$$c_{r}$$c_{0}$$c_{n}$$S_{r}$$S_{n}$$S_{m}$
Figure 9: A Sphere on ${\mathbb{R}}^{\infty}$

##  X Differentiable Structure on ${\mathbb{R}}^{\infty}$

Since each component space ${\mathbb{R}}^{n}\subset{\mathbb{R}}^{\infty}$ has
its differentiable structure, which provides a differentiable structure for
${\mathbb{R}}^{\infty}$.

###### Definition X.1

  * (i)

A hyper vector space is said to be differentiable, if all its component
subspaces are differentiable.

  * (ii)

Let $\Theta\subset{\mathbb{Z}}_{+}$, with $(\Theta,\prec)$ be a sublattice of
MD-1. Then

${\mathbb{R}}^{\Theta}:=\bigcup_{\theta\in\Theta}{\mathbb{R}}^{\theta}\subset{\mathbb{R}}^{\infty},$

as a sub hyper vector space with the inherited topological structure from
${\mathbb{R}}^{\infty}$ and the differentiable structure from
${\mathbb{R}}^{\theta}$, where $\theta\in\Theta$, is called an Euclidean hyper
manifold.

###### Remark X.2

  * (i)

It follows from the definition that the differentiability is component-wise
defined. Hence ${\mathbb{R}}^{\infty}$ is obviously differentiable. Then so is
${\mathbb{R}}^{\Theta}$.

  * (ii)

Unlike the differentiability, the topological structure of an Euclidean hyper
manifold is globally defined.

In the following, the major objects of differential geometry will be defined
over Euclidean hyper manifold.

###  X-A Smooth Functions

###### Definition X.3

A continuous function $h:{\mathbb{R}}^{\Theta}\rightarrow{\mathbb{R}}$ is said
to be $C^{r}$ (or $C^{\infty}$, or $C^{\omega}$), Denoted the set of $C^{r}$
(or $C^{\infty}$, or $C^{\omega}$, which means the set of analytic functions.)
functions by $C^{r}({\mathbb{R}}^{\Theta})$ (or
$C^{\infty}({\mathbb{R}}^{\Theta})$, or $C^{\omega}({\mathbb{R}}^{\Theta})$),
if for each $\theta\in\Theta$ the restriction $f|_{{\mathbb{R}}^{\theta}}$ is
$C^{r}$ (or $C^{\infty}$, or $C^{\omega}$).

###### Example X.4

  * (i)

Define $f:{\mathbb{R}}^{\infty}\rightarrow{\mathbb{R}}$ by

$f|_{{\mathbb{R}}^{n}}(x)=\mathop{\sum}\limits_{i=1}^{n}|x_{i}|,\quad
x\in{\mathbb{R}}^{n},\;n\in{\mathbb{Z}}_{+}.$

Then it is obvious that $f\in C^{0}({\mathbb{R}}^{\infty})$.

  * (ii)

Define $f:{\mathbb{R}}^{\infty}\rightarrow{\mathbb{R}}$ by

$f(x)=\|x\|_{{\mathcal{V}}}^{2}.$

Then

$f|_{{\mathbb{R}}^{n}}(x)=\frac{1}{n}\mathop{\sum}\limits_{i=1}^{n}x^{2}_{i},\quad
x\in{\mathbb{R}}^{n},\;n\in{\mathbb{Z}}_{+}.$

It is clear that $f\in C^{\omega}({\mathbb{R}}^{\infty})$.

###### Remark X.5

A necessary condition for $h$ to be continuous is

$\displaystyle h(x)=h(y),\quad x\leftrightarrow y.$ (260)

An alternative definition for smooth functions is as follows.

###### Definition X.6

A function $h:{\mathbb{R}}^{\Theta}\rightarrow{\mathbb{R}}$ is said to be
$C^{r}$ (or $C^{\infty}$, or $C^{\omega}$), if

  * (i)

(260) holds.

  * (ii)

For each $\theta\in\Theta$ the restriction $f|_{{\mathbb{R}}^{\theta}}$ is
$C^{r}$ (or $C^{\infty}$, or $C^{\omega}$).

Since (260) ensures the continuity among different component spaces, the
Definition X.6 is obviously equivalent to Definition X.3. Hereafter, for
statement ease, we allow $r=\infty$ or $r=\omega$.

In general, such smooth functions are hardly constructed. Particularly, in
further discussions (about vector fields etc.) and from application point of
view, we may weaken the requirement.

###### Definition X.7

Assume $\Theta$ has a lattice structure and $\Xi\subset\Theta$ is a filter. A
function $h:{\mathbb{R}}^{\Theta}\rightarrow{\mathbb{R}}$ is said to be
$C^{r}$ w.r.t. $\Xi$, denoted by $h\in C^{r}_{\Xi}(\Theta)$, if

  * (i)

(260) holds for
$x,~{}y\in{\mathbb{R}}^{\Xi}=\bigcup_{\xi\in\Xi}{\mathbb{R}}^{\xi}$.

  * (ii)

For each $\theta\in\Theta$ the restriction $f|_{{\mathbb{R}}^{\theta}}$ is
$C^{r}$ .

We need a lemma for further construction.

###### Lemma X.8

Let $u,v,w\in{\mathbb{Z}}_{+}$, where $u|v$ and $v|w$. Then

$\displaystyle\Pi^{w}_{u}=\Pi^{w}_{v}\Pi^{v}_{u}.$ (261)

Proof. Set $s=v/u$ and $t=w/v$. Then

$\Pi^{w}_{v}=\frac{1}{t}\left(I_{v}\otimes{\bf
1}^{T}_{t}\right)\left(I_{w}\otimes 1\right)=\frac{1}{t}\left(I_{v}\otimes{\bf
1}^{T}_{t}\right).$

Similarly,

$\Pi^{v}_{u}=\frac{1}{s}\left(I_{u}\otimes{\bf 1}^{T}_{s}\right).$

Then

$\begin{array}[]{l}\Pi^{w}_{v}\Pi^{v}_{u}=\frac{1}{st}\left\\{[(I_{u}\otimes{\bf
1}_{s}^{T})I_{r}]\otimes{\bf 1}^{T}_{t}\right\\}\\\
~{}=\frac{1}{st}(I_{u}\otimes{\bf 1}^{T}_{st})=\Pi^{w}_{u}.\end{array}$

$\Box$

The following example shows how to construct a $C_{\Xi}^{r}$ function from a
component space to whole hyper-manifold.

###### Example X.9

Let $h(x)\in C^{r}({\mathbb{R}}^{\theta})$. Define Define

$\Xi=\\{n\theta\;|\;n\in{\mathbb{Z}}_{+}\\}.$

Then it is clear that $(\Xi,\prec)$ is a sublattice of
$({\mathbb{Z}}_{+},\prec)$, and it is a filter. Define

$\displaystyle H(z):=\begin{cases}h(\Pi^{m}_{\theta}(z)),\quad
z\in{\mathbb{R}}^{m},\;m\in\Xi,\\\ 0,\quad m\not\in\Xi.\end{cases}$ (262)

We claim that $H(z)\in C_{\Xi}^{r}({\mathbb{R}}^{\Theta})$. Condition (ii) of
Definition X.7 is obviously true. We have only to prove (i). Let
$x\in{\mathbb{R}}^{r\theta}$, $y\in{\mathbb{R}}^{s\theta}$ and
$x\leftrightarrow y$. Denote $r\wedge s=t$, then there exists
$z\in{\mathbb{R}}^{t}$ such that $x=z\otimes{\bf 1}_{r/t}$ and $y=z\otimes{\bf
1}_{s/t}$.

Set $\Pi^{tn}_{n}(z):=z_{0}$. Using Lemma X.8, we have

$\begin{array}[]{l}H(x)=h(\Pi^{rn}_{n}x)=h(\Pi^{tn}_{n}\Pi^{rn}_{tn}(x))\\\
~{}=h(\Pi^{tn}_{n}(z))=h(z_{0}).\\\ \end{array}$

Similarly,

$H(y)=h(z_{0}).$

That is, (i) follows.

Note that instead of (262) we can also define

$\displaystyle H(z):=h(\Pi^{m}_{\theta}(z)).$ (263)

But we can still only prove (i) for $x,y\in{\mathbb{R}}^{\Xi}$, where
$\Xi=\\{kn\;|\;k\in{\mathbb{Z}}_{+}\\}$.

###  X-B Vector Fields and Covector Fields

###### Definition X.10

Consider ${\mathbb{R}}^{\Theta}$.

  * (i)

The tangent space of ${\mathbb{R}}^{\Theta}$ is defined by

$\displaystyle
T({\mathbb{R}}^{\Theta})=\bigcup_{\theta\in\Theta}T({\mathbb{R}}^{\theta}).$
(264)

  * (ii)

A vector field $f\in V^{r}({\mathbb{R}}^{\Theta})$ if it satisfied

    * –

$\displaystyle f(x)\leftrightarrow f(y),\quad x\leftrightarrow y.$ (265)

    * –

$\displaystyle f|_{{\mathbb{R}}^{\theta}}\in
V^{r}({\mathbb{R}}^{\theta}),\quad\theta\in\Theta.$ (266)

###### Example X.11

Let $f(x)\in V^{r}({\mathbb{R}}^{n})$ and $n\in\Theta$. Then the smallest
filter containing $n$ is

$\Xi:=\\{kn\;|\;k\in{\mathbb{Z}}_{+}\\}.$

It is easy to extend $f(x)$ to a vector field on
$V^{r}_{\Xi}({\mathbb{R}}^{\Theta})$. To this end, we define

$\displaystyle F(z):=\begin{cases}\Pi^{n}_{m}f(\Pi^{m}_{n}(z)),\quad m=kn,\\\
0,\quad m\neq kn,\;z\in{\mathbb{R}}^{m}.\\\ \end{cases}$ (267)

It is ready to verify that the condition (ii) of Definition X.7 is satisfied.
We prove condition (i) is also satisfied for $x,y\in{\mathbb{R}}^{\Xi}$. Let
$x\in{\mathbb{R}}^{r\theta}$, $y\in{\mathbb{R}}^{s\theta}$ and
$x\leftrightarrow y$. Denote $r\wedge s=t$, then there exists
$z\in{\mathbb{R}}^{t}$ such that $x=z\otimes{\bf 1}_{r/t}$ and $y=z\otimes{\bf
1}_{s/t}$. Then

$\begin{array}[]{l}F(x)=\Pi^{n}_{rn}f(\Pi^{tn}_{n}\Pi^{rn}_{tn}(x))\\\
~{}=\Pi^{n}_{rn}f(\Pi^{tn}_{n}(z))\\\ ~{}=\Pi^{n}_{rn}f(z_{0})\\\
~{}=(I_{n}\otimes{\bf 1}_{r})f(z_{0})\\\ ~{}=f(z_{0})\otimes
I_{r}.\end{array}$

Similarly, we have

$F(y)~{}=f(z_{0})\otimes I_{s}.$

Hence

$F(x)\leftrightarrow F(y).$

###### Definition X.12

Consider ${\mathbb{R}}^{\Theta}$.

  * (i)

The cotangent space of ${\mathbb{R}}^{\Theta}$ is defined by

$\displaystyle
T^{*}({\mathbb{R}}^{\Theta})=\bigcup_{\theta\in\Theta}T^{*}({\mathbb{R}}^{\theta}).$
(268)

  * (ii)

A co-vector field $\omega\in V^{*r}({\mathbb{R}}^{\Theta})$ if it satisfied

    * –

$\displaystyle\omega(x)\leftrightarrow\omega(y),\quad x\leftrightarrow y.$
(269)

    * –

$\displaystyle\omega|_{{\mathbb{R}}^{\theta}}\in
V^{*r}({\mathbb{R}}^{\theta}),\quad\theta\in\Theta.$ (270)

Similarly to Example X.11, we are also able to extend a co-vector field
$\omega(x)\in V^{*r}({\mathbb{R}}^{n})$ to
$V^{*r}_{\Xi}({\mathbb{R}}^{\Theta})$.

###  X-C Distribution and Co-Distribution

###### Definition X.13

Consider ${\mathbb{R}}^{\Theta}$. A distribution (co-distribution) is a rule,
which assign for each $x\in{\mathbb{R}}^{\Theta}$ a subspace $D_{x}\subset
T_{x}({\mathbb{R}}^{\Theta})$ ( $D^{*}_{x}\subset
T^{*}_{x}({\mathbb{R}}^{\Theta}))$ , satisfying

  * (i)

For each $n\in\Theta$, $D|_{{\mathbb{R}}^{n}}$ is a distribution over
$T({\mathbb{R}}^{n})$ ($D^{*}|_{{\mathbb{R}}^{n}}$ is a co-distribution over
$T^{*}({\mathbb{R}}^{n})$ );

  * (ii)

Assume $x\in{\mathbb{R}}^{m}$, $y\in{\mathbb{R}}^{n}$, $m,n\in\Theta$, and
$x\leftrightarrow y$, then $t=m\wedge n\in\Theta$ and there exists
$z\in{\mathbb{R}}^{t}$ such that $x=z\otimes{\bf 1}_{m/t}$ and $y=z\otimes{\bf
1}_{n/t}$, and then

$\displaystyle\begin{array}[]{l}D_{x}=D_{z}\otimes{\bf 1}_{m/t},\quad
D_{y}=D_{z}\otimes{\bf 1}_{n/t}.\\\ (D^{*}_{x}=D^{*}_{z}\otimes{\bf
1}^{T}_{m/t},\quad D^{*}_{y}=D^{*}z\otimes{\bf 1}^{T}_{n/t}.)\end{array}$
(273)

  * (iii)

A distribution $D$ (co-distribution $D^{*}$) is said to be $C^{r}$ if it is
generated by a set of $C^{r}$ vector fields as

$\displaystyle\begin{array}[]{l}D(x)=\mathop{\sum}\limits_{i=1}^{s}c_{i}(x)f_{i}(x),\\\
(D^{*}(x)=\mathop{\sum}\limits_{i=1}^{s}c_{i}(x)\omega_{i}(x),)\end{array}$
(276)

where $f_{i}(x)\in V^{r}({\mathbb{R}}^{\Theta})$ ($\omega_{i}(x)\in
V^{*r}({\mathbb{R}}^{\Theta})$) are $C^{r}$ vector fields (co-vector fields)
and the coefficients $c_{i}(x)\in C^{r}({\mathbb{R}}^{\Theta})$ are $C^{r}$
functions.

  * (iv)

Let $\Xi$ be a filter of $\Theta$, if $D(x)$ ($D^{*}(x)$) is only $C^{r}$ over
$x\in{\mathbb{R}}^{\Xi}$, $D(x)$ ($D^{*}(x)$) is said to be a $C^{r}$
distribution (co-distribution) w.r.t. $\Xi$, briefly, $C^{r}_{\Xi}$
($C^{*r}_{\Xi}$).

###  X-D Tensor Fields

###### Definition X.14

Consider ${\mathbb{R}}^{\Theta}$. A tensor field with covariant order $r$ and
contra-variant order $s$, denoted by ${\bf t}\in{\bf
T}^{r}_{s}({\mathbb{R}}^{\Theta})$, is a rule, which assign for each
$x\in{\mathbb{R}}^{\Theta}$ a set of multi-linear mappings, satisfying

  * (i)

For each $n\in\Theta$, for ${\bf t}|_{{\mathbb{R}}^{n}}$ is a tensor field

${\bf t}|_{{\mathbb{R}}^{n}}\in{\bf T}^{r}_{s}({\mathbb{R}}^{n}).$

  * (ii)

Assume $x\in{\mathbb{R}}^{m}$, $y\in{\mathbb{R}}^{n}$, $m,n\in\Theta$. Assume
$x\leftrightarrow y$, then $t=m\wedge n\in\Theta$ and there exists
$z\in{\mathbb{R}}^{t}$ such that $x=z\otimes{\bf 1}_{m/t}$ and $y=z\otimes{\bf
1}_{n/t}$, and then

$\displaystyle\begin{array}[]{l}M^{\bf
t}(x)=\underbrace{{\mathcal{E}}^{T}_{m/t}\otimes\cdots\otimes{\mathcal{E}}^{T}_{m/t}}_{s}\otimes
M^{\bf
t}_{z}\otimes\underbrace{{\mathcal{E}}_{m/t}\otimes\cdots\otimes{\mathcal{E}}_{m/t}}_{r},\\\
M^{\bf
t}(y)=\underbrace{{\mathcal{E}}^{T}_{n/t}\otimes\cdots\otimes{\mathcal{E}}^{T}_{n/t}}_{s}\otimes
M^{\bf
t}_{z}\otimes\underbrace{{\mathcal{E}}_{n/t}\otimes\cdots\otimes{\mathcal{E}}_{n/t}}_{r},\\\
\end{array}$ (279)

where $M^{\bf t}(p)$, $p=x,z,y$ are the structure matrix of ${\bf t}$ at
$x,y,z$ respectively.

  * (iii)

A tensor field ${\bf t}$ is said to be $C^{r}$ if the structure matrix of
${\bf t}$ is overall continuous, and within each $n\in\Theta$ , ${\bf
t}|_{{\mathbb{R}}^{n}}$ is $C^{r}$.

  * (iv)

Let $\Xi$ be a filter of $\Theta$, if the structure matrix of ${\bf t}$ is
continuous within $\Xi$ (under the subspace topology), and within each
$n\in\Theta$ , ${\bf t}|_{{\mathbb{R}}^{n}}$ is $C^{r}$, denoted by ${\bf
t}\in{{\bf T}^{r}_{s}}_{\Xi}({\mathbb{R}}^{\Theta})$.

(We refer to [19] for detailed discussion on the differential structure over
hyper-manifolds within an essentially the same topological space.)

##  XI Hyper Manifold

$U\cap V$${\bf
M}$${\mathbb{R}}^{\Theta}$${\mathbb{R}}^{\Theta}$$\Psi$$\Phi$$\Psi_{\theta}$$\Phi_{\theta}$$U_{\theta}$$V_{\theta}$$U$$V$
Figure 10: Hyper Manifold

###### Definition XI.1

Let ${\bf M}$ be a second countable topological space with a partition

${\bf M}=\bigcup_{\theta\in\Theta}{\bf M}_{\theta},$

where each ${\bf M}_{\theta}$ is a Hausdorff space (w.r.t. the inherited
topology).

Assume

${\mathcal{U}}:=\left\\{{\mathcal{U}}_{\lambda},\;|\;\lambda\in\Lambda\right\\}$

is an open cover of ${\bf M}$, such that for each $U\in{\mathcal{U}}$ there
exists a mapping $\pi:U\rightarrow U^{{\mathbb{R}}}$, where
$U^{{\mathbb{R}}}\subset{\mathbb{R}}^{\Theta}$ is an open set, and $\pi$ is a
homeomorphism.

Denote by $U_{\theta}=U\cap{\bf M}_{\theta}$, $\theta\in\Theta$. Let
$\Psi:U_{\theta}\rightarrow U^{{\mathbb{R}}}_{\theta}$ with
$\Psi:U_{\theta}\rightarrow
U^{{\mathbb{R}}}_{\theta}\subset{\mathbb{R}}^{\theta}$, and
$\Phi:V_{\theta}\rightarrow V^{{\mathbb{R}}}_{\theta}$ with
$\Phi:V_{\theta}\rightarrow
V^{{\mathbb{R}}}_{\theta}\subset{\mathbb{R}}^{\theta}$, and $U_{\theta}\cap
V_{\theta}\neq\emptyset$. Then

$\displaystyle\Phi\circ\Psi^{-1}:\Psi(U_{\theta}\bigcap
V_{\theta})\rightarrow\Phi(U_{\theta}\bigcap V_{\theta})$ (280)

is $C^{r}$.

Then ${\bf M}$ is called a $C^{r}$ hyper-manifold.

Each ${\mathcal{U}}_{\lambda}\in{\mathcal{U}}$ is called a coordinate chart.

Figure 10 depicts a hyper-manifold.

All the differentiable objects of ${\bf M}$ can be defined over each
coordinate chart $U$ (or $V$) in such a way, that the image of the objects
over $\Psi(U)$ are properly defined. Then it is also properly defined on each
$U$, and hence on ${\bf M}$.

###### Example XI.2

  * (i)

Consider ${\mathcal{M}}$. Let $\Theta=\\{m\times
n\;|\;m,n\in{\mathbb{Z}}_{+}\\}\subset Z_{+}$. Define a mapping
$\Psi:{\mathcal{M}}\rightarrow{\mathbb{R}}^{\Theta}$ as

$\displaystyle\Psi:A\mapsto V_{c}(A),$ (281)

then ${\mathbb{R}}^{\Theta}$ can be considered as a global coordinate chart of
${\mathcal{M}}$. Hence ${\mathcal{M}}$ is a hyper-manifold.

  * (ii)

Consider $\overline{\mathcal{M}}$. Let $\Theta=\\{m\times
n+1\;|\;m,n\in{\mathbb{Z}}_{+}\\}\subset Z_{+}$. Define a mapping
$\Psi:\overline{\mathcal{M}}\rightarrow{\mathbb{R}}^{\Theta}$ as

$\displaystyle\Psi:A=aI_{m\times n}+A_{0}\mapsto(a,V_{c}(A_{0})),$ (282)

then ${\mathbb{R}}^{\Theta}$ can be considered as a global coordinate chart of
$\overline{\mathcal{M}}$. Hence $\overline{\mathcal{M}}$ is a hyper-manifold.

##  XII Lie Algebra and Lie Group of Non-square Matrices

###  XII-A Lie Algebra of Non-square Matrices

###### Definition XII.1

[22]Consider ${\mathcal{M}}_{m\times n}$. The Lie bracket is defined by

$\displaystyle[A,B]_{\,\rotatebox{-90.0}{$\ltimes$}\,}:=A\,\rotatebox{-90.0}{$\ltimes$}\,B-B\,\rotatebox{-90.0}{$\ltimes$}\,A.$
(283)

###### Proposition XII.2

[22] $gl(m\times n,{\mathbb{R}}):=({\mathcal{M}}_{m\times
n},[\cdot,\cdot]_{\,\rotatebox{-90.0}{$\ltimes$}\,})$ is a Lie algebra.

Note that when $m=n$, $gl(m\times n,{\mathbb{R}})=gl(n,{\mathbb{R}})$. Hence
$gl(m\times n,{\mathbb{R}})$ is a natural generalization of the classical
general linear algebra from square case to non-square case.

To construct a Lie group $GL(m\times n,{\mathbb{R}})$ with $gl(m\times
n,{\mathbb{R}})$ as its Lie algebra, we need some preparations.

###### Definition XII.3

[45]

  * (i)

Let ${\bf g}=(G,[\cdot,\cdot])$ be a Lie algebra, and $H\subset G$ ba a
subspace. If

$\displaystyle[h_{1},h_{2}]\in H,\quad h_{1},h_{2}\in H,$ (284)

then ${\bf h}=(H,[\cdot,\cdot])$ is called a Lie sub-algebra of ${\bf g}$.

  * (ii)

Let ${\bf h}$ be a Lie sub-algebra of ${\bf g}$. If

$\displaystyle[h,g]\in H,\quad h\in H,\;g\in G,$ (285)

then ${\bf h}$ is called an ideal of ${\bf g}$.

It follows from the definition immediately that

###### Proposition XII.4

Consider $gl(m\times n,{\mathbb{R}})$.

  * (i)

Set

$\displaystyle Q_{m\times n}:=\left\\{A\in{\mathcal{M}}_{m\times
n}\;|\;Tr^{\Pi}(A)=0\right\\},$ (286)

then $Q_{m\times n}$ is an ideal of $gl(m\times n,{\mathbb{R}})$.

  * (ii)

Set

$\displaystyle Z_{m\times n}:=\left\\{A\in{\mathcal{M}}_{m\times
n}\;|\;\Pi_{A}=rI_{m},\;r\in{\mathbb{R}}\right\\},$ (287)

then $Z_{m\times n}$ is an ideal of $gl(m\times n,{\mathbb{R}})$.

###### Example XII.5

Consider $gl(2\times 3,{\mathbb{R}})$. Solving

$A\Psi_{3\times 2}=\lambda I_{2}$

yields

$\displaystyle\begin{array}[]{ccl}Z_{m\times
n}&=&\left\\{\left.\sqrt{6}\begin{bmatrix}1.5&-2&1\\\
1&-2&1.5\end{bmatrix}\lambda+(1,-2,1)\begin{bmatrix}\alpha\\\
\beta\end{bmatrix}\right|\right.\\\
{}\hfil&{}\hfil&\left.\lambda,\alpha,\beta\in{\mathbb{R}}\right\\}.\end{array}$
(290)

In general, when $m<n$, $Z_{m\times n}$ is an $(n-m)m+1$ dimensional subspace.

###### Definition XII.6

[45]

  * (i)

Let ${\bf h}=(H,[\cdot,\cdot])$ and ${\bf g}=(G,[\cdot,\cdot])$ be two Lie
algebras. If there exists a linear mapping $\pi:H\rightarrow G$ such that

$\displaystyle\pi[h_{1},h_{2}]=[\pi(h_{1}),\pi(h_{2})],\quad h_{1},h_{2}\in
H,$ (291)

then ${\bf h}$ and ${\bf g}$ are homomorphic and $\pi$ is a homomorphism.

  * (ii)

$\pi:H\rightarrow G$ is an one-to-one and onto mapping, and both $\pi$ and
$\pi^{-1}$ are homomorphisms, then ${\bf h}$ and ${\bf g}$ are isomorphic and
$\pi$ is an isomorphism.

###### Proposition XII.7

Consider $gl(m\times n,{\mathbb{R}})$ and $gl(m_{0}\times n_{0},{\mathbb{R}})$
Assume $m_{0}|m$, $n_{0}|n$, and denote $m=m_{0}a$ and $n=n_{0}b$. Define
$\pi:gl(m_{0}\times n_{0},{\mathbb{R}})\rightarrow gl(m\times n,{\mathbb{R}})$
by

$\displaystyle\pi(A):=A\otimes{\mathcal{E}}_{a\times b},\quad A\in
gl(m_{0}\times n_{0}).$ (292)

Then $\pi$ is a homomorphism.

Proof. We have to show that

$[\pi(A_{0}),\pi(B_{0})]=\pi\left([A_{0},B_{0}]\right),\quad A_{0},B_{0}\in
gl(m_{0}\times n_{0},{\mathbb{R}}).$

It suffices to show that setting $A=A_{0}\otimes{\mathcal{E}}_{a\times b}$ and
$B=B_{0}\otimes{\mathcal{E}}_{a\times b}$, then

$\displaystyle
A\,\rotatebox{-90.0}{$\ltimes$}\,B=\pi(A_{0}\,\rotatebox{-90.0}{$\ltimes$}\,B_{0}),\quad
A_{0},B_{0}\in gl(m_{0}\times n_{0},{\mathbb{R}}).$ (293)

Let $t_{0}=m_{0}\vee n_{0}$ and $\xi=m\vee n$. Then

$\begin{array}[]{l}\pi(A_{0}\,\rotatebox{-90.0}{$\ltimes$}\,B_{0})=\pi\left[(A\otimes{\mathcal{E}}^{T}_{t_{0}/n_{0}})(B\otimes{\mathcal{E}}_{t_{0}/m_{0}})\right]\\\
~{}=\left[(A\otimes{\mathcal{E}}^{T}_{t_{0}/n_{0}})(B\otimes{\mathcal{E}}_{t_{0}/m_{0}})\right]\otimes{\mathcal{E}}_{a\times
b}\\\
~{}=\left[(A\otimes{\mathcal{E}}^{T}_{t_{0}/n_{0}})(B\otimes{\mathcal{E}}_{t_{0}/m_{0}})\right]\otimes({\mathcal{E}}_{a}E^{T}_{b})\\\
~{}=(A\otimes{\mathcal{E}}_{a\times
t_{0}/n_{0}})(B\otimes{\mathcal{E}}_{t_{0}/m_{0}\times b}).\\\ \end{array}$

And

$\begin{array}[]{l}A\,\rotatebox{-90.0}{$\ltimes$}\,B=(A_{0}\otimes{\mathcal{E}}_{a\times
b})\,\rotatebox{-90.0}{$\ltimes$}\,(B_{0}\otimes{\mathcal{E}}_{a\times b})\\\
~{}=(A_{0}\otimes{\mathcal{E}}_{a\times
b}\otimes{\mathcal{E}}^{T}_{\xi/n})(B_{0}\otimes{\mathcal{E}}_{a\times
b}\otimes{\mathcal{E}}_{\xi/m})\\\ ~{}=(A_{0}\otimes{\mathcal{E}}_{a\times
b\xi/n})(B_{0}\otimes{\mathcal{E}}_{a\xi/m\times b})\\\
~{}=(A_{0}\otimes{\mathcal{E}}_{a\times\xi/n_{0}})(B_{0}\otimes{\mathcal{E}}_{\xi/m_{0}\times
b})\\\ ~{}=(A\otimes{\mathcal{E}}_{a\times
t_{0}/n_{0}})(B\otimes{\mathcal{E}}_{t_{0}/m_{0}\times b}).\\\ \end{array}$

The conclusion follows.

$\Box$

###### Corollary XII.8

Assume $gl(m\times n,{\mathbb{R}})$, $gl(m_{0}\times n_{0},{\mathbb{R}})$ and
the mapping $\pi$ are as in Proposition XII.7. Then $Im(\pi)$ is a Lie sub-
algebra of $gl(m\times n,{\mathbb{R}})$.

Proof. Note that $\pi$ is a homomorphism. Moreover, it is obvious that $\pi$
is one to one, then the conclusion is obvious.

$\Box$

###  XII-B Lie Group of Non-square Matrices

Define $I_{m\times n}$ as a formal identity for the algebra
$({\mathcal{M}}_{m\times n}\,\rotatebox{-90.0}{$\ltimes$}\,)$, which satisfies

$\displaystyle\begin{array}[]{l}I_{m\times
n}\,\rotatebox{-90.0}{$\ltimes$}\,I_{m\times n}=I_{m\times n}\\\ I_{m\times
n}\,\rotatebox{-90.0}{$\ltimes$}\,A=A\,\rotatebox{-90.0}{$\ltimes$}\,I_{m\times
n}=A,\quad\forall A\in{\mathcal{M}}_{m\times n}.\end{array}$ (296)

Then we have an extended matrix-ring as

$\displaystyle\overline{\mathcal{M}}_{m\times n}:=\left\\{aI_{m\times
n}+A_{0}\;|\;a\in{\mathbb{R}},~{}A_{0}\in{\mathcal{M}}_{m\times n}\right\\},$
(297)

with

$\displaystyle\begin{array}[]{l}(aI_{m\times n}+A_{0})+(bI_{m\times
n}+B):=(a+b)I_{m\times n}\\\ ~{}+(A_{0}+B_{0}),\\\ (aI_{m\times
n}+A_{0})\,\rotatebox{-90.0}{$\ltimes$}\,(bI_{m\times
n}+B_{0}):=(ab)I_{m\times n}+aB_{0}\\\
~{}+bA_{0}+A_{0}\,\rotatebox{-90.0}{$\ltimes$}\,B_{0},~{}a,b\in{\mathbb{R}},~{}A,B\in{\mathcal{M}}_{m\times
n}.\end{array}$ (302)

Note that where $m\neq n$ the objects $\\{aI_{m\times n}+A_{0}\\}$ of an
extended matrix-ring are not matrices unless $a=0$.

Let $A\in{\mathcal{M}}_{m\times n}$. Then $\Pi_{A}=A\Psi_{n\times m}$, where
$\Psi_{n\times m}$ defined by (42), is called the restricted (square) form of
$A$, because

$A\,\rotatebox{-90.0}{$\ltimes$}\,x=\Pi_{A}x,\quad\forall
x\in{\mathbb{R}}^{m}.$

Define

$\displaystyle A^{<k>}=\begin{cases}I_{m\times n},\quad k=0,\\\
\underbrace{A\,\rotatebox{-90.0}{$\ltimes$}\,\cdots\,\rotatebox{-90.0}{$\ltimes$}\,A}_{k},\quad
k>0.\\\ \end{cases}$ (303)

In addition, let $p(x)=x^{n}+c_{n-1}x^{n-1}+\cdots+c_{1}x+c_{0}$, we define

$\displaystyle\begin{array}[]{l}p<A>:=A^{<n>}+c_{n-1}A^{<n-1>}+\cdots+c_{1}A\\\
+c_{0}I_{m\times n}\in\overline{\mathcal{M}}_{m\times n}.\end{array}$ (306)

Then we have the following result.

###### Theorem XII.9

(Generalized Cayley-Hamilton theorem)[22] Let
$p(x)=x^{m}+c_{m-1}x^{m-1}+\cdots+c_{1}x+c_{0}$ be the characteristic function
of $\Pi_{A}$, which is also called the characteristic function of $A$. Then

$\displaystyle A^{<m+1>}+c_{m-1}A^{<m>}+\cdots+c_{1}A^{<2>}+c_{0}A=0.$ (307)

###### Remark XII.10

  * (i)

When $n<m$ we can take the characteristic function of $\Pi_{A^{T}}$ as the
characteristic function of $A$. This alternative choice can reduce the degree
of the characteristic function.

  * (ii)

Formally, we may rewritten (307) into the form as

$\displaystyle A^{<m>}+c_{m-1}A^{<m-1>}+\cdots+c_{1}A+c_{0}I^{A}_{m\times
n}=0.$ (308)

Then if $c_{0}\neq 0$ we can define

$\displaystyle I^{A}_{m\times
n}:=-\frac{1}{c_{0}}\left[A^{<m>}+c_{m-1}A^{<m-1>}+\cdots+c_{1}A\right].$
(309)

According to Generalized Cayley-Hamilton theorem, it is clear that

$\displaystyle A\,\rotatebox{-90.0}{$\ltimes$}\,I^{A}_{m\times
n}=I^{A}_{m\times n}\,\rotatebox{-90.0}{$\ltimes$}\,A=A.$ (310)

Moreover, for any polynomial $p<A>$, we also have

$\displaystyle(p<A>)\,\rotatebox{-90.0}{$\ltimes$}\,I^{A}_{m\times
n}=I^{A}_{m\times n}\,\rotatebox{-90.0}{$\ltimes$}\,(p<A>)=p<A>.$ (311)

Then $A$ is said to be relatively invertible and define its relative inverse
by

$\displaystyle\begin{array}[]{l}A^{<-1>}=-\frac{1}{c_{0}}\left[A^{<m-1>}+c_{m-1}A^{<m-2>}+\cdots\right.\\\
~{}\left.+c_{1}I^{A}_{m\times n}\right].\end{array}$ (314)

It follows that

$\displaystyle
A\,\rotatebox{-90.0}{$\ltimes$}\,A^{<-1>}=A^{<-1>}\,\rotatebox{-90.0}{$\ltimes$}\,A=I^{A}_{m\times
n}.$ (315)

It seems that if $A$ is relatively invertible, then $I_{m\times n}|_{p<A>}$ is
$I^{A}_{m\times n}$. Unfortunately, $I^{A}_{m\times n}$ depends on $A$.
Moreover, it is a matrix while $I_{m\times n}$ is not.

###### Definition XII.11

$A\in\overline{\mathcal{M}}_{m\times n}$ is invertible, if there exists
$B\in\overline{\mathcal{M}}_{m\times n}$, such that

$A\,\rotatebox{-90.0}{$\ltimes$}\,B=B\,\rotatebox{-90.0}{$\ltimes$}\,A=I_{m\times
n}.$

###### Remark XII.12

  * (i)

Assume $A=aI_{m\times n}+A_{0}$, where $A_{0}\in{\mathcal{M}}_{m\times n}$. It
is obvious that a necessary condition for $A$ being invertible is $a\neq 0$.

  * (ii)

If $A=aI_{m\times n}+A_{0}$ is invertible and $A^{<-1>}=B$. Then
$A^{\prime}=\frac{1}{a}A$ is also invertible and $(A^{\prime})^{<-1>}=aB$.
Hence, to find the inverse of $A$, we can,without loss of generality, assume
$a=1$.

  * (iii)

Assume $A=I_{m\times n}+A_{0}$, $A$ is invertible, if and only if, there
exists $B=I_{m\times n}+B_{0}$, where $B_{0}\in{\mathcal{M}}_{m\times n}$ such
that

$\displaystyle\begin{array}[]{l}A\,\rotatebox{-90.0}{$\ltimes$}\,B=(I_{m\times
n}+A_{0})\,\rotatebox{-90.0}{$\ltimes$}\,(I_{m\times n}+B_{0})\\\ =I_{m\times
n}+A_{0}+B_{0}+A_{0}\,\rotatebox{-90.0}{$\ltimes$}\,B_{0}.\end{array}$ (318)

  * (iv)

Assume $m=n$. Then $\overline{\mathcal{M}}_{n\times n}={\mathcal{M}}_{m\times
n}$ because now $I_{n\times n}=I_{n}\in{\mathcal{M}}_{n\times n}$. Then it is
easily verifiable that $A\in{\mathcal{M}}_{n\times n}$ is invertible (under
the classical definition), if and only if, (318) holds. Hence Definition
XII.11 can be considered as an extension of the classical invertibility to
non-square case.

  * (v)

The $GL(m\times n,{\mathbb{R}})$ has been defined by [22] as

$\displaystyle\begin{array}[]{l}GL(m\times
n,{\mathbb{R}}):=\left\\{A=I_{m\times
n}+A_{0}\;|\;A_{0}\in{\mathcal{M}}_{m\times n}\right.\\\
~{}\left.\mbox{and}~{}A~{}\mbox{is invertible}\right\\}.\end{array}$ (321)

Then it was proved in [22] that $GL(m\times n,{\mathbb{R}})$ is a Lie group,
which has $gl(m\times n,{\mathbb{R}})$ as its Lie algebra.

Next, we consider when $A=I_{m\times n}+A_{0}\in\overline{\mathcal{M}}$ is
invertible. Assume the characteristic function of $A_{0}$ is

$\displaystyle p(x)=x^{k-1}+c_{k-1}x^{k-2}+\cdots+c_{2}x+c_{1}.$ (322)

Using generalized Cayley-Hamilton theorem, we have

$A_{0}^{<k>}+c_{k-1}A_{0}^{<k-1>}+\cdots+c_{2}A_{0}^{<2>}+c_{1}A_{0}=0.$

We assume

$\displaystyle
B_{0}=x_{k-1}A_{0}^{<k-1>}+x_{k-2}A_{0}^{<k-2>}+\cdots+x_{2}A_{0}^{<2>}+x_{1}A_{0}.$
(323)

Then

$\begin{array}[]{l}A_{0}\,\rotatebox{-90.0}{$\ltimes$}\,B_{0}=x_{k-1}A_{0}^{<k>}+x_{k-2}A_{0}^{<k-1>}+\cdots\\\
~{}+x_{2}A_{0}^{<3>}+x_{1}A^{<2>}_{0}\\\
~{}=-x_{k-1}(c_{k-1}A_{0}^{<k-1>}+\cdots+c_{2}A_{0}^{<2>}+c_{1}A_{0})\\\
~{}+x_{k-2}A_{0}^{<k-1>}+\cdots+x_{2}A_{0}^{<3>}+x_{1}A^{<2>}_{0}\\\
\end{array}$

Setting

$\displaystyle A_{0}+B_{0}+A_{0}\,\rotatebox{-90.0}{$\ltimes$}\,B_{0}=0$ (324)

and comparing the coefficients yield the following linear system

$\displaystyle\begin{cases}-c_{k-1}x_{k-1}+x_{k-2}+x_{k-1}=0\\\
-c_{k-2}x_{k-1}+x_{k-3}+x_{k-2}=0\\\ \cdots\\\ -c_{2}x_{k-1}+x_{1}+x_{2}=0\\\
-c_{1}x_{k-1}+x_{1}+1=0.\end{cases}$ (325)

Solving (325) yields

$\displaystyle\begin{cases}x_{1}=c_{1}x_{k-1}-1,\\\
x_{2}=(c_{2}-c_{1})x_{k-1}+1,\\\ x_{3}=(c_{3}-c_{2}+c_{1})x_{k-1}-1,\\\
\cdots,\\\
x_{k-2}=\left(c_{k-2}-c_{k-3}\pm\cdots+(-1)^{k-1}c_{1}\right)x_{k-1}+(-1)^{k-2},\\\
\left(c_{k-1}-c_{k-2}+c_{k-3}\pm\cdots+(-1)^{k}c_{1}-1\right)x_{k-1}\\\
~{}+(-1)^{k-1}=0.\end{cases}$ (326)

Then we have the following result.

###### Theorem XII.13

Consider $A=I_{m\times n}+A_{0}$ and assume the characteristic function of
$A_{0}$ is (322). Then $A$ is invertible, if and only if,

$\displaystyle c_{k-1}-c_{k-2}+c_{k-3}\pm\cdots+(-1)^{k-2}c_{1}-1\neq 0.$
(327)

Proof. The previous constructing process is a constructive proof for the
sufficiency. As for the necessity, it is reasonable to assume the inverse of
$A$ is an analytic function of $A_{0}$. Taking the generalized Cayley-Hamilton
theorem into consideration, it is obvious that the form of $B_{0}$, expressed
by (323) is a general expression of an analytic function of $A_{0}$.

$\Box$

###### Remark XII.14

  * (i)

If (327) holds, (326) with (323) provide the inverse of $A$ immediately.

  * (ii)

It is easy to see that

$\displaystyle I_{m\times n}=\exp(0_{m\times n}).$ (328)

Then by continuity, one sees easily that there exists an open neighborhood
${\mathcal{O}}$ of $0_{m\times n}$, say

$0_{m\times n}\in{\mathcal{O}}\subset{\mathbb{R}}^{mn},$

such that for each $B\in{\mathcal{O}}$ the charactoristic function

$p_{B}(x)=x^{k-1}+c_{k-1}x^{k-2}+\cdots+c_{2}x+c_{1},$

satisfies

$\left|c_{k-1}-c_{k-2}+c_{k-3}\pm\cdots+(-1)^{k-2}c_{1}\right|<1.$

Hence

$I_{m\times n}+B\in GL(m\times n,{\mathbb{R}}),\quad B\in{\mathcal{O}}.$

  * (iii)

By possibly shrinking ${\mathcal{O}}$, we can assume

$\displaystyle\ln{<I_{m\times
n}+B>}=\mathop{\sum}\limits_{i=1}^{\infty}(-1)^{i-1}B^{<i>},\quad
B\in{\mathcal{O}}$ (329)

converges. Then we define

$\displaystyle GL_{0}(m\times n,{\mathbb{R}}):=\left\\{I_{m\times
n}+B\;|\;B\in{\mathcal{O}}\right\\}.$ (330)

It follows that

$\displaystyle GL_{0}(m\times n,{\mathbb{R}})\subset\exp({\mathcal{O}}),$
(331)

where $\exp({\mathcal{O}})$ is an open sub-manifold of $GL(m\times
n,{\mathbb{R}})$, which shows that $GL_{0}(m\times n,{\mathbb{R}})$ itself is
a sub-Lie group of $GL(m\times n,{\mathbb{R}})$.

###### Example XII.15

Consider $A=I_{2\times 3}+A_{0}$, where
$A_{0}=(a_{i,j})\in{\mathcal{M}}_{2\times 3}$. For statement ease, we let
$\Psi_{3\times 2}$ be as in (44). Then we have

  * (i)

$\begin{array}[]{l}\Pi_{A}=A_{0}\Psi_{3\times 2}=A_{0}\begin{bmatrix}2&0\\\
1&1\\\ 0&2\end{bmatrix}\\\
~{}=\begin{bmatrix}2a_{1,1}+a_{1,2}&a_{1,2}+2a_{1,3}\\\
2a_{2,1}+a_{2,2}&a_{2,2}+2a_{2,3}\end{bmatrix}.\end{array}$

The characteristic function of $A_{0}$ is

$p(x)=x^{2}+c_{2}x+c1,$

where

$\begin{array}[]{l}c_{2}=-(2a_{1,1}+a_{1,2}+a_{2,2}+2a_{2.3}),\\\
c_{1}=(2a_{1,1}+a_{1,2})(a_{2,2}+2a_{2,3})\\\
~{}-(2a_{2,1}+a_{2,2})(a_{1,2}+2a_{1,3}).\end{array}$

We conclude that as long as $c_{2}-c_{1}-1\neq 0$, $A$ is invertible. The
inverse is

$B_{0}=x_{2}A_{0}^{<2>}+x_{1}A_{0},$

where

$x_{1}=\frac{c_{2}-1}{c_{1}-c_{2}+1},\quad x_{2}=\frac{1}{c_{1}-c_{2}+1}.$

  * (ii)

To verify the above result, we calculate the inverse by definition. Assume
$B_{0}=(b_{i,j})\in{\mathcal{M}}_{2\times 3}$ such that

$\displaystyle A_{0}+B_{0}+A_{0}\,\rotatebox{-90.0}{$\ltimes$}\,B_{0}=0.$
(332)

A direct calculation shows that

$A_{0}\,\rotatebox{-90.0}{$\ltimes$}\,B_{0}:=C_{0}=(c_{i,j})\in{\mathcal{M}}_{2\times
3},$

where

$\begin{array}[]{l}c_{1,1}=2a_{1,1}b_{1,1}+a_{1,2}b_{1,1}+a_{1,2}b_{2,1}+2a_{1,3}b_{2,1},\\\
c_{1,2}=2a_{1,1}b_{1,2}+a_{1,2}b_{1,2}+a_{1,2}b_{2,2}+2a_{1,3}b_{2,2},\\\
c_{1,3}=2a_{1,1}b_{1,3}+a_{1,2}b_{1,3}+a_{1,2}b_{2,3}+2a_{1,3}b_{2,3},\\\
c_{2,1}=2a_{2,1}b_{1,1}+a_{2,2}b_{1,1}+a_{2,2}b_{2,1}+2a_{2,3}b_{2,1},\\\
c_{2,2}=2a_{2,1}b_{1,2}+a_{2,2}b_{1,2}+a_{2,2}b_{2,2}+2a_{2,3}b_{2,2},\\\
c_{2,3}=2a_{2,1}b_{1,3}+a_{2,2}b_{1,3}+a_{2,2}b_{2,3}+2a_{2,3}b_{2,3}.\\\
\end{array}$

(332) leads to a linear system as

$\displaystyle\begin{bmatrix}\alpha&0&0&\beta&0&0\\\ 0&\alpha&0&0&\beta&0\\\
0&0&\alpha&0&0&\beta\\\ \gamma&0&0&\delta&0&0\\\ 0&\gamma&0&0&\delta&0\\\
0&0&\gamma&0&0&\delta\\\ \end{bmatrix}\begin{bmatrix}b_{1,1}\\\ b_{1,2}\\\
b_{1,3}\\\ b_{2,1}\\\ b_{2,2}\\\ b_{2,3}\\\
\end{bmatrix}=\begin{bmatrix}-a_{1,1}\\\ -a_{1,2}\\\ -a_{1,3}\\\ -a_{2,1}\\\
-a_{2,2}\\\ -a_{2,3}\\\ \end{bmatrix},$ (333)

where

$\begin{array}[]{l}\alpha=2a_{1,1}+a_{1,2}+1,\\\ \beta=a_{1,2}+2a_{1,3},\\\
\gamma=2a_{2,1}+a_{2,2},\\\ \delta=2a_{2,3}+a_{2,2}+1.\\\ \end{array}$

Then it is clear that (333) has unique solution, if and only if,

$\alpha\delta-\beta\gamma\neq 0.$

A careful calculation shows that

$\alpha\delta-\beta\gamma=-(c_{2}-c1-1).$

That is, the condition obtained in (i) is necessary and sufficient.

In general, the following is obvious.

###### Proposition XII.16

Let

$\displaystyle{\mathcal{T}}_{m\times
n}:=\left(\\{A\in\overline{\mathcal{M}}_{m\times n}\;|\;A~{}\mbox{is
invertible}\\},\,\rotatebox{-90.0}{$\ltimes$}\,\right).$ (334)

Then ${\mathcal{T}}_{m\times n}$ is a group.

###### Definition XII.17

Let $B\in{\mathcal{M}}_{m\times n}$.

  * (i)

$\displaystyle\begin{array}[]{l}GL(m\times
n,{\mathbb{R}}):=\left\\{A=I_{m\times n}+A_{0}\right.\\\
~{}\left.\;|\;A_{0}\in{\mathcal{M}}_{m\times
n}~{}\mbox{and}~{}A\in{\mathcal{T}}_{m\times n}\right\\}.\end{array}$ (337)

called the $m\times n$ general linear group.

  * (iI)

Let $B\in{\mathcal{M}}_{m\times n}$. Define

$\displaystyle\exp{<B>}:=I_{m\times
n}+\mathop{\sum}\limits_{n=1}^{\infty}\frac{1}{n!}B^{<n>},$ (338)

Then

$\exp{<B>}\in GL(m\times n,{\mathbb{R}}).$

###### Theorem XII.18

[22] $GL(m\times n,{\mathbb{R}})$ is a Lie group, which has $gl(m\times n)$ as
its Lie algebra.

###### Remark XII.19

  * (i)

$GL(m\times n,{\mathbb{R}})<{\mathcal{T}}_{m\times n}.$

  * (ii)

Define

$\displaystyle{\mathcal{B}}:=\\{\,\rotatebox{-90.0}{$\ltimes$}\,_{i=1}^{k}\exp(B_{i})\;|\;B_{i}\in{\mathcal{M}}_{m\times
n},\;k<\infty\\}.$ (339)

Then ${\mathcal{B}}$ is a group and

${\mathcal{B}}<GL(m\times n,{\mathbb{R}})$

is a sub-Lie group, which is a path-connected component of $GL(m\times
n,{\mathbb{R}})$.

  * (iii)

$A_{0}$ can be considered as the coordinate of $A=I_{m\times n}+A_{0}$. It is
then clear that

$\displaystyle\dim(GL(m\times n,{\mathbb{R}}))=mn.$ (340)

  * (iv)

$\overline{\mathcal{M}}_{m\times n}$ is a ring, which is also an $mn+1$
dimensional vector space. It can also be naturally embedded into
${\mathbb{R}}^{mn+1}$ by

$A=aI_{m\times
n}+A_{0}\mapsto(a,A_{0}^{1,1},A_{0}^{1,2},\cdots,A_{0}^{m,n})^{T}\in{\mathbb{R}}^{mn+1}.$

Hence $\overline{\mathcal{M}}_{m\times n}$ can be naturally considered as a
topological vector space, which has the classical topology of Euclidian space
${\mathbb{R}}^{mn+1}$. Now $gl(m\times n,{\mathbb{R}})$ can be considered as
its $mn$ dimensional subspace, and $GL(m\times n,{\mathbb{R}})$ is also its
$m\times n$ dimensional sub-manifold. When $m\neq n$ $gl(m\times n)\bigcap
GL(m\times n)=\emptyset$. This is depicted in Fig.11.

  * (v)

It is clear that

$I_{m\times n}=\exp(0_{m\times n}),$

which is the identity of $GL(m\times n,{\mathbb{R}})$. So it is not an element
in $gl(m\times n,{\mathbb{R}})$. When we extend it to be the identity of the
ring $\overline{\mathcal{M}}$, we have the relations shown in (296).

$gl(m\times n,{\mathbb{R}})$$GL(m\times
n,{\mathbb{R}})$$\exp$$\overline{\mathcal{M}}_{m\times n}$ Figure 11:
$gl(m\times n,{\mathbb{R}})$ and $GL(m\times n,{\mathbb{R}})$

###  XII-C Relationship with General Linear Group/Algebra

As an immediate consequence of Adi’s theorem, we have the following theorem
[45].

###### Theorem XII.20

Let ${\bf g}$ be a Lie algebra over ${\mathbb{R}}$. Then there exists an
integer $n\geq 1$ and an analytic subgroup $G$ of $GL(n,{\mathbb{R}})$ such
that the Lie algebra of $G$ is isomorphic to ${\bf g}$.

According to Theorem XII.20, there is a Lie sub-algebra of ${\bf g}\subset
gl(k,{\mathbb{R}})$, such that $gl(m\times n,{\mathbb{R}})$ is isomorphic to
${\bf g}<gl(k,{\mathbb{R}})$. This motivates the following result.

###### Proposition XII.21

Consider $\Box:{\mathcal{M}}_{m\times n}\rightarrow{\mathcal{M}}_{t\times t}$,
where $t=m\vee n$. Then $\Box:gl(m\times n,{\mathbb{R}})\rightarrow
gl(t,{\mathbb{R}})$ is a Lie algebra homomorphism. Moreover, $Im(\Box)$ is a
sub-algebra of $gl(t,{\mathbb{R}})$.

Proof. Let $A,B\in{\mathcal{M}}_{m\times n}$. It suffices to show that

$\displaystyle\Box(A\,\rotatebox{-90.0}{$\ltimes$}\,B)=\Box(A)\Box(B),\quad
A,B\in{\mathcal{M}}_{m\times n}.$ (341)

$\begin{array}[]{l}\Box(A\,\rotatebox{-90.0}{$\ltimes$}\,B)=\Box\left((A\otimes{\mathcal{E}}^{T}_{t/n})(B\otimes{\mathcal{E}}_{t/m}\right)\\\
~{}=\left((A\otimes{\mathcal{E}}^{T}_{t/n})(B\otimes{\mathcal{E}}_{t/m}\right)\otimes{\mathcal{E}}_{t/m\times
t/n}\\\
~{}=\left((A\otimes{\mathcal{E}}^{T}_{t/n})(B\otimes{\mathcal{E}}_{t/m}\right)\otimes\left({\mathcal{E}}_{t/m}{\mathcal{E}}^{T}_{t/n}\right)\\\
~{}=\left((A\otimes{\mathcal{E}}^{T}_{t/n}\otimes{\mathcal{E}}_{t/m}\right)\left(B\otimes{\mathcal{E}}_{t/m}\otimes{\mathcal{E}}^{T}_{t/n}\right)\\\
~{}=\left((A\otimes{\mathcal{E}}_{t/m\times
t/n})(B\otimes{\mathcal{E}}_{t/m\times t/n}\right)\\\ ~{}=\Box(A)\Box(B).\\\
\end{array}$

Hence $\Box:gl(m\times n,{\mathbb{R}})\rightarrow gl(t,{\mathbb{R}})$ is a Lie
algebra homomorphism. Now it is obvious that $\Box$ is one-to-one, then

$Im(\Box)=\Box({\mathcal{M}}_{m\times n})$

is a Lie sub-algebra of $gl(t,{\mathbb{R}})$.

$\Box$

Define

$\displaystyle\begin{array}[]{l}GL(Im({\mathcal{M}}_{m\times
n}),{\mathbb{R}}):=\left\\{\prod_{i=1}^{k}\exp(B_{i})\;|\;\right.\\\
~{}\left.B_{i}\in\Box({\mathcal{M}}_{m\times
n}),\;k<\infty\right\\}.\end{array}$ (344)

Then

$\displaystyle GL(Im({\mathcal{M}}_{m\times
n}),{\mathbb{R}})<GL(n,{\mathbb{R}})$ (345)

is a Lie sub-group.

Moreover, if we choose

$\displaystyle\begin{array}[]{l}GL(m\times
n,{\mathbb{R}}):=\left\\{\,\rotatebox{-90.0}{$\ltimes$}\,_{i=1}^{k}\exp(<B_{i}>)\;|\;\right.\\\
~{}\left.B_{i}\in gl_{m\times
n},{\mathbb{R}}),\;k<\infty\right\\},\end{array}$ (348)

then

$\displaystyle GL(m\times n,{\mathbb{R}})\cong GL(Im({\mathcal{M}}_{m\times
n}),{\mathbb{R}})$ (349)

is a Lie group isomorphism.

##  XIII From Topological Hyper Group to Hyper Lie Group

###  XIII-A Topological Hyper Group

###### Definition XIII.1

Assume ${\mathcal{G}}=(G,*,{\bf e})$ is both a hyper group and a topological
space. If both the product and the inverse are continuous, then
${\mathcal{G}}$ is called a topological hyper group.

###### Example XIII.2

  * (i)

Consider ${\mathcal{T}}=\\{A\in{\mathcal{M}}_{1}\;|\;A~{}\mbox{is
invertible}\\}$. Then it is easy to verify that $({\mathcal{T}},\ltimes,{\bf
e})$ (or $({\mathcal{T}},\rtimes,{\bf e})$ ) is a topological hyper group,
where ${\bf e}=\\{I_{n}\;|\;n\in{\mathbb{Z}}_{+}\\}$.

  * (ii)

Consider the permutation hyper group ${\bf S}$. By identifying $\sigma\in{\bf
S}_{n}$ with its permutation matrix $M_{\sigma}\in{\mathcal{P}}_{n}$ as
$\pi_{n}:{\bf S}_{n}\rightarrow{\mathcal{P}}_{n}$. Then we can define
$\pi:{\bf S}\rightarrow{\mathcal{P}}$ by

$\displaystyle\pi|_{{\bf S}_{n}}=\pi_{n},\quad n\in{\mathbb{Z}}_{+}.$ (350)

It is easy to see that $\pi$ is a bijective mapping. Then we define
$\circ_{\ell}:{\bf S}\times{\bf S}\rightarrow{\bf S}$ and $\circ_{r}:{\bf
S}\times{\bf S}\rightarrow{\bf S}$ as

$\displaystyle\begin{array}[]{l}\sigma\circ_{\ell}\mu=\pi^{-1}\left[M_{\sigma}\ltimes
M_{\mu}\right],\\\ \sigma\circ_{r}\mu=\pi^{-1}\left[M_{\sigma}\rtimes
M_{\mu}\right].\\\ \end{array}$ (353)

  * (iii)

Consider $({\mathcal{T}},\ltimes,{\bf e})$ (or $({\mathcal{T}},\rtimes,{\bf
e})$), which is a topological hyper group. As the hyper-subgroup of
$({\mathcal{T}},\ltimes,{\bf e})$ (or $({\mathcal{T}},\rtimes,{\bf e})$), the
$({\mathcal{P}},\ltimes,{\bf e})$ (or $({\mathcal{P}},\rtimes,{\bf e})$) with
its inherited topology is a topological hyper group.

  * (iv)

Using the topology of $({\mathcal{P}},\ltimes,{\bf e})$ (or
$({\mathcal{P}},\rtimes,{\bf e})$), $({\bf S},\circ_{\ell},{\bf Id})$ (or
$({\bf S},\circ_{r},{\bf Id})$) is a topological hyper group.

###  XIII-B Hyper Lie Group

###### Definition XIII.3

Assume ${\mathcal{G}}=(G,*,{\bf e})$ is a topological hyper group and at the
same time it is also an analytic hyper-manifold with both the product and the
inverse as analytic mappings. Then ${\mathcal{G}}$ is called a hyper Lie
group.

###### Example XIII.4

Consider ${\mathcal{G}}:=({\mathbb{R}}^{\infty},\hat{+},{\bf e})$, where ${\bf
e}=\\{0_{m\times n}\;|\;m,n\in{\mathbb{Z}}_{+}\\}$. Equipped by its Euclidean
hyper-manifold structure, ${\mathcal{G}}$ is a hyper Lie group.

##  XIV Hyper Linear Algebra

###  XIV-A Matrix Hyper Lie Algebra

###### Definition XIV.1

Consider ${\mathcal{M}}_{0}\subset{\mathcal{M}}$. Assume there are two binary
operators:
$+:{\mathcal{M}}_{0}\times{\mathcal{M}}_{0}\rightarrow{\mathcal{M}}_{0}$,
$\times:{\mathcal{M}}_{0}\times{\mathcal{M}}_{0}\rightarrow{\mathcal{M}}_{0}$,
such that $({\mathcal{M}}_{0},+)$ is a hyper-vector space;
$({\mathcal{M}}_{0},+,\times)$ is a hyper-ring, then
$({\mathcal{M}}_{0},+,\times)$ is called a matrix hyper-algebra.

###### Proposition XIV.2

Assume $({\mathcal{M}}_{0},+,\times)$ a matrix hyper-algebra. A Lie bracket is
defined by

$\displaystyle[A,B]:=A\times B-B\times A.$ (354)

Then it satisfies the following conditions for Lie algebra.

  * (i)

Skew Symmetry:

$\displaystyle[A,B]=-[B,A],\quad A,B\in{\mathcal{M}}_{0}.$ (355)

  * (ii)

Bi-Linearity:

$\displaystyle[aA+bB,C]=a[A,C]+b[B,C],\quad A.B.C\in{\mathcal{M}}_{0}.$ (356)

  * (iii)

Jacobi Identity:

$\displaystyle[A,[B,C]]+[B,[C,A]]+[C[A,B]]=0.$ (357)

That is, a matrix hyper-ring with a Lie bracket defined by (354) is called a
hyper-Lie algebra.

Proof. (355) comes from definition (354) immediately. The distributivity of
hyper-ring leads to (356). (357) follows from a straightforward computation,
using the linearity. $\Box$

A matrix hyper-algebra over matrices with Lie bracket defined by (354) is
called a hyper linear algebra.

Moreover, the following consequence is obvious.

###### Proposition XIV.3

Assume $({\mathcal{M}}_{0},+,\times)$ is a hyper linear algebra, and the
identity set of $({\mathcal{M}}_{0},+)$ is ${\bf e}$. Then

  * (i)

each vector space ${\mathcal{M}}_{0}^{e}$, $e\in{\bf e}$, is a general linear
algebra;

  * (ii)

if $+$ and $\times$ are consistent w.r.t. the equivalence, the equivalence
classes $({\mathcal{M}}_{0}/\sim,+,\times)$ is a Lie algebra, called the
equivalence linear algebra.

It is well known that $({\mathcal{M}}_{n\times n}+,\times)$ with the Lie
bracket defined by (354) is the general linear algebra $gl(n,{\mathbb{R}})$.
So the hyper linear algebra is a natural generalization of the classical
general linear algebra.

The above argument shows that as long as a matrix hyper-ring exists, it leads
to a hyper linear algebra automatically. Hence, we have the following hyper
matrix Lie algebras based on the matrix hyper-rings[19]:

###### Example XIV.4

  * (i)

Consider $({\mathcal{M}}_{1},\bar{+},\ltimes)$, defined in Remark VII.2
equation (207). Define

$\displaystyle[A,B]:=A\ltimes B\bar{-}B\ltimes A.$ (358)

Then $({\mathcal{M}}_{1},\bar{+},\ltimes)$ with the Lie bracket defined by
(358) is a hyper-Lie algebra. Denote this hyper-Lie algebra by
$gl_{s}^{1}({\mathbb{R}})$.

  * (ii)

Consider $({\mathcal{M}}_{1},\hat{+},\circ)$, defined in Remark VII.2 equation
(208). Define

$\displaystyle[A,B]:=A\circ B\hat{-}B\circ A.$ (359)

Then $({\mathcal{M}}_{1},\hat{+},\circ)$ with the Lie bracket defined by (359)
is a hyper-Lie algebra. Denote this hyper-Lie algebra by
$gl_{s}^{2}({\mathbb{R}})$.

In the following we construct the most general hyper linear algebra over the
set of all matrices.

###  XIV-B Hyper Linear Algebra $gl({\mathbb{R}})$

###### Definition XIV.5

Consider

$\displaystyle
gl({\mathbb{R}}):=({\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}).$
(360)

It is already known that $gl({\mathbb{R}})$ is a hyper-ring. Define

$\displaystyle[A,B]_{\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}}:=A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B\hat{-}B\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A.$
(361)

Then $gl({\mathbb{R}})$ becomes a hyper linear algebra.

Recall that the identity set is

${\bf e}=\left\\{0_{m\times n}\;|\;m,n\in{\mathbb{Z}}_{+}\right\\}.$

First, for each $e=0_{m\times n}\in{\bf e}$
${\mathcal{M}}_{e}={\mathcal{M}}_{m\times n}$. Hence we have

###### Proposition XIV.6

  * (i)

$\displaystyle gl(m\times n,{\mathbb{R}}):=\left(({\mathcal{M}}_{m\times
n},+,\,\rotatebox{-90.0}{$\ltimes$}\,),[\cdot,\cdot]\right)$ (362)

is a Lie algebra.

  * (ii)

Particularly,

$\displaystyle gl(n,{\mathbb{R}}):=\left(({\mathcal{M}}_{n\times
n},+,\times),[\cdot,\cdot]\right)$ (363)

is a Lie algebra, which is the well known $n$-th order general linear algebra.

  * (iii)

$\displaystyle
gl({\mathbb{R}})=\bigcup_{m=1}^{\infty}\bigcup_{n=1}^{\infty}gl(m\times
n,{\mathbb{R}})$ (364)

is a partition of the hyper-general linear algebra. And each Lie-algebra
$gl(m\times n,{\mathbb{R}})$ is called its component Lie-algebra.

Then $A\sim B$ means there exist $0_{a\times b}$ and $0_{c\times d}$ such that

$\displaystyle A\hat{+}0_{a\times b}=B\hat{+}0_{c\times d}.$ (365)

Assume $A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times q}$,
$m\vee p=r$, and $n\wedge q=s$. By deleting common facts the (362) can be
rewritten as

$\displaystyle A\hat{+}0_{r/m\times s/n}=B\hat{+}0_{r/p\times s/q}.$ (366)

It leads to

$\displaystyle A\otimes{\mathcal{E}}_{r/m\times
s/n}=B\otimes{\mathcal{E}}_{r/p\times s/q}.$ (367)

###### Lemma XIV.7

Consider
$gl({\mathbb{R}}):=({\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$.
The equivalence is consistent with $\hat{+}$ and
$\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$.

Proof.

The consistence of the equivalence w.r.t. $\hat{+}$ has already been proved in
Proposition III.14. We prove the consistence of equivalence w.r.t.
$\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$.

Assume $A_{1}\in{\mathcal{M}}_{m\times n}$, $A_{2}\in{\mathcal{M}}_{p\times
q}$, $B_{1}\in{\mathcal{M}}_{\alpha\times\beta}$,
$B_{2}\in{\mathcal{M}}_{\gamma\times\delta}$, and

$\begin{array}[]{c}m\vee\alpha=a,~{}n\vee\beta=b,~{}p\vee\gamma=c,\\\
q\vee\delta=d,~{}m\wedge p=s,~{}n\wedge q=t,\\\
\alpha\wedge\gamma=\xi,~{}\beta\wedge\delta=\eta,~{}\xi\vee t=u,\\\ a\vee
b=\theta=ux,~{}c\vee d=\lambda=uy,\\\
a=a_{0}s,~{}b-b_{0}\eta,~{}c=c_{0}s,~{}d=d_{0}\eta.\end{array}$

Assume $A_{1}\sim A_{2}$ and $B_{1}\sim B_{2}$, we have to show that

$\displaystyle A_{1}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{1}\sim
A_{2}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{2}.$ (368)

We use equivalent condition (367) for matrix equivalence. Using Lemma III.16,
there exist $A_{0}\in{\mathcal{M}}_{s\times t}$ and
$B_{0}\in{\mathcal{M}}_{\xi\times\eta}$, such that

$\begin{array}[]{cc}A_{1}=A_{0}\otimes{\mathcal{E}}_{m/s\times
n/t},&A_{2}=A_{0}\otimes{\mathcal{E}}_{p/s\times q/t},\\\
B_{1}=B_{0}\otimes{\mathcal{E}}_{\alpha/\xi\times\beta/\eta},&B_{2}=B_{0}\otimes{\mathcal{E}}_{\gamma/\xi\times\delta/\eta}.\\\
\end{array}$

Then we have

$\begin{array}[]{l}A_{1}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{1}=\left(A_{1}\otimes{\mathcal{E}}{a/m\times
b/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(B_{1}\otimes{\mathcal{E}}{a/\alpha\times
b/\beta}\right)\\\ ~{}=\left(A_{0}\otimes{\mathcal{E}}_{m/s\times
n/t}\otimes{\mathcal{E}}_{a/m\times b/n}\right)\\\
~{}\,\rotatebox{-90.0}{$\ltimes$}\,\left(B_{0}\otimes{\mathcal{E}}_{\alpha/\xi\times\beta/\eta}\otimes{\mathcal{E}}_{a/\alpha\times
b/\beta}\right)\\\ ~{}=\left(A_{0}\otimes{\mathcal{E}}_{a/s\times
b/t}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(B_{0}\otimes{\mathcal{E}}_{a/\xi\times
b/\eta}\right)\\\ ~{}=\left(A_{0}\otimes{\mathcal{E}}_{a/s\times
b/t}\otimes{\mathcal{E}}^{T}_{\theta/b}\right)\left(B_{0}\otimes{\mathcal{E}}_{a/\xi\times
b/\eta}\otimes{\mathcal{E}}_{\theta/a}\right)\\\
~{}=\left(A_{0}\otimes{\mathcal{E}}_{a/s\times\theta/t}\right)\left(B_{0}\otimes{\mathcal{E}}_{\theta/\xi\times
b/\eta}\right)\\\ ~{}=\left(A_{0}\otimes{\mathcal{E}}_{a_{0}\times
u/t}\right)\left(B_{0}\otimes{\mathcal{E}}_{u/\xi\times b_{0}}\right)\\\
~{}=\left[\left(A_{0}\otimes{\mathcal{E}}_{1\times
u/t}\right)\left(B_{0}\otimes{\mathcal{E}}_{u/\xi\times
1}\right)\right]\otimes\left({\mathcal{E}}_{a_{0}}{\mathcal{E}}^{T}_{b_{0}}\right).\\\
~{}=\left[\left(A_{0}\otimes{\mathcal{E}}_{1\times
u/t}\right)\left(B_{0}\otimes{\mathcal{E}}_{u/\xi\times
1}\right)\right]\otimes\left({\mathcal{E}}_{a_{0}\times b_{0}}\right).\\\
\end{array}$

Similarly, we also have

$A_{2}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{2}=\left[\left(A_{0}\otimes{\mathcal{E}}_{1\times
u/t}\right)\left(B_{0}\otimes{\mathcal{E}}_{u/\xi\times
1}\right)\right]\otimes\left({\mathcal{E}}_{c_{0}\times d_{0}}\right).\\\ $

We conclude that

$A_{1}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{1}\sim
A_{2}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{2}.$

$\Box$

Denote the quotient space by

$\Sigma={\mathcal{M}}/\sim,$

and using Lemma XIV.7, the following result is obvious.

###### Proposition XIV.8

The equivalent space
$gl_{\Sigma}({\mathbb{R}}):=(\Sigma,\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$
is a Lie algebra.

###### Definition XIV.9

Assume $V$ is a hyper vector space and there is a Lie bracket
$[\cdot,\cdot]:V\times V\rightarrow V$, such that $(V,[\cdot,\cdot])$ is a Lie
algebra.

  * (i)

If $N\subset V$ is a hyper vector subspace and $(N,[\cdot,\cdot])$ is also a
Lie bracket, then $(N,[\cdot,\cdot])$ is called a hyper Lie sub-algebra of
$(V,[\cdot,\cdot])$.

  * (ii)

If $N\subset V$ is a hyper Lie sub-algebra, and

$\displaystyle[A,N]\subset N,\quad\forall A\in V,$ (369)

then $N$ is called an ideal of $V$.

As an immediate consequence, the following result is straightforward
verifiable.

###### Proposition XIV.10

$({\mathcal{M}}_{\mu},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$ is a
hyper Lie sub-algebra of $gl({\mathbb{R}})$, denoted by
$gl_{\mu}({\mathbb{R}})$.

###### Remark XIV.11

Summarizing the matrix hyper Lie algebras we obtained so far, we have the
followings:

  * (i)

$({\mathcal{M}}_{1},\bar{+},\ltimes)$, its corresponding Lie hyper algebra is
$gl_{s}^{1}({\mathbb{R}})$ described in Example XIV.4.

  * (ii)

$({\mathcal{M}}_{1},\hat{+},\circ)$, its corresponding Lie hyper algebra is
$gl_{s}^{2}({\mathbb{R}})$ described in Example XIV.4.

  * (iii)

$({\mathcal{M}},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$, its
corresponding Lie hyper algebra is $gl({\mathbb{R}})$ described in Definition
XIV.5.

  * (iv)

$({\mathcal{M}}_{\mu},\hat{+},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$, its
corresponding Lie hyper algebra is $gl_{\mu}({\mathbb{R}})$, which is a Lie
sub-hyper algebra of $gl({\mathbb{R}})$.

The following example presents an ideal of $gl({\mathbb{R}})$.

###### Example XIV.12

Define

$\displaystyle T_{\Box}:=\\{A\in{\mathcal{M}}\;|\;Tr(\Box(A))=0\\}.$ (370)

Then $T_{\Box}$ is an ideal of $gl({\mathbb{R}})$.

To see this, assume $A\in{\mathcal{M}}_{m\times n}$,
$B\in{\mathcal{M}}_{p\times q}$, and $m\vee n\vee p\vee q=r$. Then a
straightforward computation shows that

$\begin{array}[]{l}[A,B]_{\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}}=\left(A\otimes{\mathcal{E}}_{r/m\times
r/n}\right)\left(B\otimes{\mathcal{E}}_{r/p\times r/q}\right)\\\
-\left(B\otimes{\mathcal{E}}_{r/p\times
r/q}\right)\left(A\otimes{\mathcal{E}}_{r/m\times r/n}\right).\end{array}$

Hence,

$Tr([A,B]_{\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}})=0.$

The conclusion is obvious.

###  XIV-C $M$-Depending Sub Hyper Algebra

This subsection considers a kind of sub-hyper algebras of $gl({\mathbb{R}})$,
which depend on any $M\in{\mathcal{M}}$. Given $M\in{\mathcal{M}}$, we define

$\displaystyle
G_{M}:=\\{A\in{\mathcal{M}}\;|\;A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}M\hat{+}M\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{T}=0\\}.$
(371)

###### Theorem XIV.13

Set

$\displaystyle
gl(M,{\mathbb{R}}):=(G_{M},[\cdot,\cdot]_{\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}}).$
(372)

Then $gl(M,{\mathbb{R}})$ is a hyper Lie subalgebra of $gl({\mathbb{R}})$.

To prove this theorem, we need some preparations.

###### Lemma XIV.14

$\displaystyle
A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}M\hat{+}M\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{T}=0,$
(373)

if and only if,

$\displaystyle\Box(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}M)=-\Box(M\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{T}).$
(374)

Proof. Assume $M\in{\mathcal{M}}_{m\times n}$, $A\in{\mathcal{M}}_{p\times
q}$, and

$\begin{array}[]{c}m\vee p=\alpha,~{}n\vee q=\beta,\\\ m\vee q=a,~{}n\vee
p=b,\\\ \alpha\vee a=m\vee p\vee q=c,~{}\beta\vee b=n\vee p\vee q=d,\\\
\alpha\vee\beta=a\vee b=m\vee n\vee p\vee q=\xi.\end{array}$

Then

$\begin{array}[]{l}A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}M=\left(A\otimes{\mathcal{E}}_{\alpha/p\times\beta/q}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(M\otimes{\mathcal{E}}_{\alpha/m\times\beta/n}\right)\\\
~{}=\left(A\otimes{\mathcal{E}}_{\alpha/p\times\xi/q}\right)\left(M\otimes{\mathcal{E}}_{\xi/m\times\beta/n}\right)\in{\mathcal{M}}_{\alpha\times\beta}.\end{array}$
$\begin{array}[]{l}M\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{T}=\left(M\otimes{\mathcal{E}}_{a/m\times
b/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(A^{T}\otimes{\mathcal{E}}_{a/q\times
b/p}\right)\\\
~{}=\left(M\otimes{\mathcal{E}}_{a/m\times\xi/n}\right)\left(A^{T}\otimes{\mathcal{E}}_{\xi/q\times
b/p}\right)\in{\mathcal{M}}_{a\times b}.\end{array}$
$\begin{array}[]{l}A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}M\hat{+}M\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{T}=\left(A\otimes{\mathcal{E}}_{\alpha/p\times\xi/q}\right)\left(M\otimes{\mathcal{E}}_{\xi/m\times\beta/n}\right)\hat{+}\\\
~{}\left(M\otimes{\mathcal{E}}_{a/m\times\xi/n}\right)\left(A^{T}\otimes{\mathcal{E}}_{\xi/q\times
b/p}\right)\\\
~{}=\left[\left(A\otimes{\mathcal{E}}_{\alpha/p\times\xi/q}\right)\left(M\otimes{\mathcal{E}}_{\xi/m\times\beta/n}\right)\right]\otimes{\mathcal{E}}_{c/\alpha\times
d/\beta}+\\\
~{}\left[\left(M\otimes{\mathcal{E}}_{a/m\times\xi/n}\right)\left(A^{T}\otimes{\mathcal{E}}_{\xi/q\times
b/p}\right)\right]\otimes{\mathcal{E}}_{c/a\times d/b}\\\
~{}=\left(A\otimes{\mathcal{E}}_{c/p\times\xi/q}\right)\left(M\otimes{\mathcal{E}}_{\xi/m\times
d/n}\right)+\\\
~{}\left(M\otimes{\mathcal{E}}_{c/m\times\xi/n}\right)\left(A^{T}\otimes{\mathcal{E}}_{\xi/q\times
d/p}\right)\\\ \end{array}$

Plugging into (373) yields

$\displaystyle\begin{array}[]{l}\left(A\otimes{\mathcal{E}}_{c/p\times\xi/q}\right)\left(M\otimes{\mathcal{E}}_{\xi/m\times
d/n}\right)+\\\
~{}\left(M\otimes{\mathcal{E}}_{c/m\times\xi/n}\right)\left(A^{T}\otimes{\mathcal{E}}_{\xi/q\times
d/p}\right)=0_{c\times d}.\\\ \end{array}$ (377)

It is clear that (377) is equivalent to (373).

Right $\otimes$-multiplying both sides of (377) by
${\mathcal{E}}_{\xi/c\times\xi/d}$, we have

$\displaystyle\begin{array}[]{l}\left[\left(A\otimes{\mathcal{E}}_{c/p\times\xi/q}\right)\left(M\otimes{\mathcal{E}}_{\xi/m\times
d/n}\right)\right]\otimes{\mathcal{E}}_{\xi/c\times\xi/d}+\\\
~{}\left[\left(M\otimes{\mathcal{E}}_{c/m\times\xi/n}\right)\left(A^{T}\otimes{\mathcal{E}}_{\xi/q\times
d/p}\right)\right]\otimes{\mathcal{E}}_{\xi/c\times\xi/d}\\\
~{}=\left(A\otimes{\mathcal{E}}_{\xi/p\times\xi/q}\right)\left(M\otimes{\mathcal{E}}_{\xi/m\times\xi/n}\right)+\\\
~{}\left(M\otimes{\mathcal{E}}_{\xi/m\times\xi/n}\right)\left(A^{T}\otimes{\mathcal{E}}_{\xi/q\times
xi/p}\right)\\\
~{}=\Box\left(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}M\right)+\Box\left(M\otimes
A^{T}\right)=0_{\xi\times\xi}.\end{array}$ (383)

Conversely, by deleting the factor ${\mathcal{E}}_{\xi/c\times\xi/d}$ from
both sides of (383), (which is obviously “legal” as long as
${\mathcal{E}}_{\xi/c\times\xi/d}\neq 0$), we have (377). $\Box$

###### Lemma XIV.15

$\displaystyle\Box(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)=\Box{A}\circ\Box(B),$
(384)

where $\circ$ is defined by (19).

Proof. Assume $A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times
q}$, and

$\begin{array}[]{c}m\vee p=\alpha,~{}n\vee q=\beta,\\\ m\vee n=a,~{}p\vee
q=b,\\\ \alpha\vee\beta=a\vee b=m\vee n\vee p\vee q=\xi.\end{array}$

Then

$\begin{array}[]{l}\Box(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)=\Box\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\beta/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(B\otimes{\mathcal{E}}_{\alpha/p\times\beta/q}\right)\right]\\\
~{}=\Box\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\xi/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(B\otimes{\mathcal{E}}_{\xi/p\times\beta/q}\right)\right]\\\
~{}=\left[\left(A\otimes{\mathcal{E}}_{\alpha/m\times\xi/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(B\otimes{\mathcal{E}}_{\xi/p\times\beta/q}\right)\right]\otimes{\mathcal{E}}_{\xi/\alpha\times\xi/\beta}\\\
~{}=\left(A\otimes{\mathcal{E}}_{\xi/m\times\xi/n}\right)\left(B\otimes{\mathcal{E}}_{\xi/p\times\xi/q}\right)\\\
~{}=\left[\left(A\otimes{\mathcal{E}}_{a/m\times a/n}\right)\otimes
J_{\xi/a}\right]\left[\left(B\otimes{\mathcal{E}}_{b/p\times
b/q}\right)\otimes J_{\xi/b}\right]\\\ ~{}=\Box(A)\circ\Box(B).\end{array}$

$\Box$

As an immediate consequence, we have

$\displaystyle\Box\left({\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}}_{i=1}^{n}A_{i}\right)=\circ_{i=1}^{n}\Box(A_{i}),\quad
A_{i}\in{\mathcal{M}}.$ (385)

Proof of Theorem XIV.13 Assume $A,B\in G_{M}$, using Lemma XIV.14, (373) is
equivalent to

$\displaystyle\Box\left[\left([A,B]_{\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}M\right)\right]+\Box\left[M\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}[A,B]^{T}_{\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}}\right]=0.$
(386)

Using Lemma XIV.15, we have

$\begin{array}[]{l}\Box\left[\left([A,B]_{\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}M\right)\right]+\Box\left[M\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}[A,B]^{T}_{\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}}\right]\\\
~{}=\Box(A)\circ\Box(B)\circ\Box(M)-\Box(B)\circ\Box(A)\circ\Box(M)\\\
~{}+\Box(M)\circ\Box(B^{T})\circ\Box(A^{T})-\Box(M)\circ\Box(A^{T})\circ\Box(B^{T})\\\
~{}=-\Box(A)\circ\Box(M)\circ\Box(B^{T})+\Box(B)\circ\Box(M)\circ\Box(A^{T})\\\
~{}+\Box(M)\circ\Box(B^{T})\circ\Box(A^{T})-\Box(M)\circ\Box(A^{T})\circ\Box(B^{T})\\\
~{}=\Box(M)\circ\Box(A^{T})\circ\Box(B^{T})-\Box(M)\circ\Box(B^{T})\circ\Box(A^{T})\\\
~{}+\Box(M)\circ\Box(B^{T})\circ\Box(A^{T})-\Box(M)\circ\Box(A^{T})\circ\Box(B^{T})\\\
=0.\end{array}$

$\Box$

##  XV Hyper Linear Group As Hyper Lie Group

###  XV-A From Matrix Hyper Rings to Matrix Hyper Group

Observing that we have two well defined Lie group of matrices. One is well
known general linear group $GL(n,{\mathbb{R}})$ and the other one is
$GL(m\times n,{\mathbb{R}})$, which has been discussed in Section XII. It
seems that the way to construct them can be described as follows: Starting
from matrix ring, we first construct the Lie algebra by defining Lie bracket
in a natural way, then build their corresponding group called the Lie group.
Precisely, we have

$\begin{array}[]{l}({\mathcal{M}}_{n\times n},+,\times)\Rightarrow
gl(n,{\mathbb{R}})\Rightarrow GL(n,{\mathbb{R}});\\\ ({\mathcal{M}}_{m\times
n},+,\,\rotatebox{-90.0}{$\ltimes$}\,)\Rightarrow gl(m\times
n,{\mathbb{R}})\Rightarrow GL(m\times n,{\mathbb{R}}).\\\ \end{array}$

Recall Remark XIV.11, we have 4 kinds of matrix hyper-Lie algebras. This
section is aimed to build their corresponding hyper-Lie groups.

Following Remark XIV.11, we consider all 4 different matrix hyper-Lie
algebras.

  1. (1)

Consider case (i), which has been discussed in [19]. But we reconsider them
from a new point of view: Consider them as hyper groups and then pose an
${\mathbb{R}}^{\infty}$ differentiable structure on it.

Let ${\mathcal{T}}=\bigcup_{n=1}^{\infty}{\mathcal{T}}_{n}$, where
${\mathcal{T}}_{n}$ is the set of $n\times n$ nonsingular matrices. Then

$\displaystyle GL_{s}({\mathbb{R}}):=({\mathcal{T}},\ltimes,{\bf e}),$ (387)

where ${\bf e}=\\{I_{n}\;|\;n\in{\mathbb{Z}}_{+}\\}$. Then the following facts
are obvious;

     * •

$GL_{s}({\mathbb{R}})$ is a hyper group.

     * •

Embedding $M_{n\times n}$ into ${\mathbb{R}}^{n^{2}}$ by $A\mapsto V_{c}(A)$,
and using the topology ${\mathcal{T}}_{d}$, then $GL_{s}({\mathbb{R}})$
becomes a topological space and a topological hyper group, with the inherited
topology from ${\mathbb{R}}^{\infty}$.

     * •

Its differentiable structure comes from ${\mathbb{R}}^{\Xi}$ with

$\begin{array}[]{l}\Xi=\\{n=m^{2}\in{\mathbb{Z}}_{+},~{}\mbox{and}\\\
~{}~{}n=s^{2}\prec m=t^{2}~{}\mbox{if and only if}~{}s\prec t\\}.\end{array}$

  1. (2)

The topological structure in (i) can hardly be used for this case, because
$A\otimes{\bf 1}_{k\times k}$ is no longer invertible, even if $A$ is. We are
not interested in constructing a hyper-Lie group with its hyper-Lie algebra
$gl_{s}^{2}({\mathbb{R}})$.

  1. (3)

Denote by

$\begin{array}[]{l}G_{m\times n}:=\left(\\{I_{m\times
n}+A_{0}\;|\;A_{0}\in{\mathcal{M}}_{m\times n},~{}\mbox{and}\right.\\\
~{}\left.I_{m\times n}+A_{0}~{}\mbox{is
invertible.}\\},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}\right).\end{array}$

We define

$\displaystyle G_{M}:=\bigcup_{m=1}^{\infty}\bigcup_{n=1}^{\infty}G_{m\times
n}\subset\overline{\mathcal{M}}.$ (388)

First we need to show that $(G_{M},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$ is
a hyper group. Set

$\displaystyle{\bf e}:=\left\\{I_{m\times
n}\;|\;(m,n)\in{\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}\right\\},$ (389)

with the lattice structure as MD-2 lattice. We need the following definition.

###### Definition XV.1

  * (i)

The product within ${\bf e}$:

$\displaystyle I_{m\times n}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}I_{p\times
q}:=I_{(m\vee p)\times(n\vee q)}.$ (390)

  * (ii)

The product of ${\bf e}$ with ${\mathcal{M}}$: Let
$A_{0}\in{\mathcal{M}}_{p\times q}$, $m\vee p=s$, and $n\vee q=t$. Then

$\displaystyle I_{m\times
n}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A_{0}=A_{0}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}I_{m\times
n}:=A_{0}\otimes{\mathcal{E}}_{s/m\times t/n}.$ (391)

###### Definition XV.2

Let $A,~{}B\in G_{M}$, say, $A=I_{m\times n}+A_{0}$, $B=I_{p\times q}+B_{0}$,
where $A_{0}\in{\mathcal{M}}_{m\times n}$, $B_{0}\in{\mathcal{M}}_{p\times
q}$, $m\vee p=s$, and $n\vee q=t$. Then

$\displaystyle\begin{array}[]{l}A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B=I_{s\times
t}+A_{0}\otimes{\mathcal{E}}_{s/m\times
t/n}+B_{0}\otimes{\mathcal{E}}_{s/p\times t/q}\\\
+A_{0}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B_{0}\in{\mathcal{M}}_{s\times
t}.\end{array}$ (394)

###### Proposition XV.3

$(G_{M},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,},{\bf e})$ is a hyper group,
where $\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$ is defined by (388) and ${\bf
e}$ is defined by (389).

Proof.

First, we prove the associativity. By the definition of
$\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$, It is enough to show that

$\displaystyle(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C=A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}(B\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C),\quad
A,B,C\in{\mathcal{M}}\bigcup{\bf e}.$ (395)

Consider the following several cases.

  * (i)

All $A,B,C\in{\mathcal{M}}$:

Then (395) has been proved in Proposition II.28.

  * (ii)

There are one or two factors from ${\bf e}$.

We prove one case, where $A,C\in{\mathcal{M}}$ and $B\in{\bf e}$. The proofs
for all other cases are similar.

Assume $A\in{\mathcal{M}}_{m\times n}$, $B=I_{p\times q}$,
$C\in{\mathcal{M}}_{r\times s}$, and

$\begin{array}[]{l}m\vee p=a,~{}n\vee q=b,~{}a\vee r=c,\\\ b\vee s=d,~{}c\vee
d=e,~{}n\vee r=\xi,\\\ p\vee r=u,~{}q\vee s=v,~{}m\vee u=m\vee p\vee r=c,\\\
n\vee v=n\vee q\vee s=d.\end{array}$

Then

$\begin{array}[]{l}\left(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}I_{p\times
q}\right)\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C\\\
~{}=\left(A\otimes{\mathcal{E}}_{a/m\times
b/n}\right)\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C\\\
~{}=\left(A\otimes{\mathcal{E}}_{a/m\times b/n}\otimes{\mathcal{E}}_{c/a\times
d/b}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{c/r\times
d/s}\right)\\\ ~{}=\left(A\otimes{\mathcal{E}}_{c/m\times
d/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{c/r\times
d/s}\right)\\\ ~{}=\left(A\otimes{\mathcal{E}}_{c/m\times
d/n}\otimes{\mathcal{E}}^{T}_{e/d}\right)\left(C\otimes{\mathcal{E}}_{c/r\times
d/s}\otimes{\mathcal{E}}_{e/c}\right)\\\
~{}=\left(A\otimes{\mathcal{E}}_{c/m\times
e/n}\right)\left(C\otimes{\mathcal{E}}_{e/r\times d/s}\right)\\\
~{}=\left(A\otimes{\mathcal{E}}_{c/m\times\xi/n}\right)\left(C\otimes{\mathcal{E}}_{\xi/r\times
d/s}\right).\\\ \end{array}$

Similarly, we have

$\begin{array}[]{l}A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}\left(I_{p\times
q}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C\right)\\\
A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}\left(C\otimes{\mathcal{E}}_{u/r\times
v/s}\right)\\\ ~{}=\left(A\otimes{\mathcal{E}}_{c/m\times
d/n}\right)\,\rotatebox{-90.0}{$\ltimes$}\,\left(C\otimes{\mathcal{E}}_{u/r\times
v/s}\otimes{\mathcal{E}}_{c/u\times d/v}\right)\\\
~{}=\left(A\otimes{\mathcal{E}}_{c/m\times
d/n}\otimes{\mathcal{E}}^{T}_{e/d}\right)\left(C\otimes{\mathcal{E}}_{c/r\times
d/s}\otimes{\mathcal{E}}_{e/c}\right)\\\
~{}=\left(A\otimes{\mathcal{E}}_{c/m\times
e/n}\right)\left(C\otimes{\mathcal{E}}_{e/r\times d/s}\right)\\\
~{}=\left(A\otimes{\mathcal{E}}_{c/m\times\xi/n}\right)\left(C\otimes{\mathcal{E}}_{\xi/r\times
d/s}\right).\\\ \end{array}$

  * (iii)

All $A,B,C\in{\bf e}$: Let $A=I_{m\times n}$, $B=I_{p\times q}$, and
$C=I_{r\times s}$. Then a straightforward computation shows that

$(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C=A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}(B\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}C)=I_{(m\vee
p\vee r)\times(n\vee q\vee s)}.$

The associativity is proved.

Next, we have to show that if $A\in{\mathcal{M}}_{m\times n}$ and
$B\in{\mathcal{M}}_{p\times q}$ are invertible, then so is
$A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B$. Using associativity, we have

$\begin{array}[]{l}(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}(B^{-1}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{-1})\\\
~{}=A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}(B\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B^{-1})\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{-1}\\\
~{}=A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}(I_{p\times
q})\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{-1}\\\ ~{}=I_{p\times
q}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{-1})\\\
~{}=I_{p\times q}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}I_{m\times
n})=I_{(m\vee p)\times(n\vee q)}.\end{array}$

Similarly, we have

$(B^{-1}\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{-1})\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}(A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B)=I_{(m\vee
p)\times(n\vee q)}.$

Hence, $A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B$ is also invertible.

Finally, for each $e=I_{m\times n}$

$(G_{M})_{m\times n}=G_{m\times n},$

which is a group. We conclude that
$(G_{M},\hat{\,\rotatebox{-90.0}{$\ltimes$}\,})$ is a hyper group. $\Box$

###### Remark XV.4

  * (i)

The invertibility of a matrix is the most important concept in group structure
of $G_{M}$. We emphasize that the Definition XII.11 is consistent w.r.t.
classical one for square matrix. Let $A\in{\mathcal{M}}_{n\times n}$. Then we
can express

$A=I_{n\times n}+A_{0},$

where $A_{0}=A-I_{n}$. Then it is easy to see that $A$ is invertible in
classical sense, i.e., there exists $B=A^{-1}$ such that $AB=BA=I_{n}$, if and
only if,

$B=I_{n\times n}+B_{0},$

where $B_{0}=B-I_{n}$ such that

$A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}B=I_{n\times
n}+A_{0}+B_{0}+A_{0}B_{0}=I_{n\times n}.$

This shows that $I_{n\times n}=I_{n}$ and the two invertibilities are
equivalent.

  * (ii)

Assume $A=I_{m\times n}+A_{0}\in\overline{\mathcal{M}}_{m\times n}$ is
invertible, where $m\neq n$. Then $A^{T}=I_{n\times m}+A^{T}_{0}$ is also
invertible. $A\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}A^{T}$ is also invertible.
Intuitively, this seems unusual. This fact shows that the product
“$\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$” is different from the classical
matrix product, and hence when $m\neq n$ the invertibility is different from
the classical one.

###  XV-B From Matrix Hyper Group to Hyper Linear Group

$gl(m\times n,{\mathbb{R}})$$GL(m\times
n,{\mathbb{R}})$$\exp$$\overline{\mathcal{M}}_{m\times
n}$$\overline{\mathcal{M}}_{s\times t}$$\overline{\mathcal{M}}_{i\times
j}$${\mathbb{R}}^{st+1}$${\mathbb{R}}^{mn+1}$${\mathbb{R}}{ij+1}$$\Psi$$\overline{\mathcal{M}}$${\mathbb{R}}^{\Xi}$
Figure 12: Matrix Hyper-Lie Group

To put topological and differentiable structure on $G_{M}$ we have to give it
coordinates. To this end, we first construct an Euclidian hyper manifold,
${\mathbb{R}}^{\Xi}$, where

$\Xi=\\{m\times n+1,\;|\;(m,n)\in{\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}\\},$

and $\\{(m,n)\in{\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}\\}$ is the MD-2
lattice.

Then we define a mapping
$\Psi:\overline{\mathcal{M}}\rightarrow{\mathbb{R}}^{\Xi}$ by

$aI_{m\times
n}+A_{0}\mapsto(a,V_{c}(A_{0}))\in{\mathbb{R}}^{mn+1},\quad(m,n)\in{\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}.$

This mapping is described in Figure 12. Then for each $m\times n+1\in\Xi$ we
have

$\Psi|_{\overline{\mathcal{M}}_{m\times n}}:\overline{\mathcal{M}}_{m\times
n}\rightarrow{\mathbb{R}}^{mn+1}.$

And $V_{c}(A_{0})$ is taken as the coordinate frame of $GL(m\times
n,{\mathbb{R}})$. This assignment makes each $GL(m\times n,{\mathbb{R}})$ a
manifold of dimension $mn$. As a consequence, $GL({\mathbb{R}})$ becomes a
matrix hyper-Lie group, called the hyper linear group.

Then each slice $GL(m\times n,{\mathbb{R}})$ has $gl(m\times n.{\mathbb{R}})$
as its Lie algebra.

##  XVI Concluding Remarks

###  XVI-A A Mathematics over MDMVs

To investigate linear (control) systems the classical matrix theory plays a
fundamental rule. While the modern mathematics provides a solid foundation for
the matrix theory. Particularly, the concepts from abstract algebra such as
group, ring, and module are used to describe the matrices/vectors under
certain operators; the geometric structures such as vector space, metric,
inner product, topology, manifold, etc. are essential for describing the state
space of the systems. Finally the Lie group and Lie algebra structure over
matrices leads to the general linear group and general linear algebra, which
reveal the inside structure of the dynamics of linear (control) systems.

Recently, the STPs/STAs have been developed to overcome the dimension barrier
of classical matrix theory. As the cross-dimensional operators, STPs/STAs
demonstrate their dimension-free characteristic, which makes the classical
algebraic and geometric structures unsuitable for MVMDs. They, therefore, are
stimulating a new matrix theory to merge. This new matrix theory demands a new
mathematical foundation. The purpose of this paper is to explore a new
foundation, which is called the CDM. The CDM consists of three parts: hyper
algebra, hyper geometry, and hyper linear algebra and hyper linear group.

The hyper algebra consists of three new concepts: the hyper group, hyper ring,
and hyper module. The hyper group extends the classical group to a group with
multiple identities, each identity represents a component group. The cross-
dimensional operator joints all component groups together. The equivalence
group shows the unified properties of component groups. Based on hyper group,
the hyper ring and hyper module are also proposed in a natural way. They have
similar structure: component objects and equivalence rings/modules.

The hyper geometry uses the hyper group structure to construct a hyper-vector
space, which has multiple zeros. Then the hyper inner product space and hyper
metric space are also obtained. Furthermore, the metric-based topological
structure and a differentiable structure have been posed on
${\mathbb{R}}^{\infty}$, and using such topological and differentiable
structure, the hyper manifold is constructed. These objects provide a
geometric structure for MVMDs.

Finally, the Lie algebra and Lie group structure for hyper matrix group w.r.t.
$\hat{+}$ and $\hat{\,\rotatebox{-90.0}{$\ltimes$}\,}$ respectively, is
proposed. The topological hyper group is also constructed. They eventually
lead to the hyper-Lie algebra and the hyper-Lie group for MVMDs, called the
hyper general linear algebra and hyper general linear group, are established.

###  XVI-B Potential Applications

Dimension-varying systems appear in nature and from engineering systems. For
instance, In a genetic regulatory network, cells may die or birth at any time;
in the internet or some other service-based networks, some users may join in
or withdraw out from time to time; some mechanical systems, such as docking,
undocking, departure, and joining of spacecrafts [50, 32], modeling of
biological systems [49, 30], uncertain model identification [47], etc.

There are also many dimension uncertain systems. For instance, a single
electric power generator may be described by 2-dimensional model, or
3-dimensional, or even 5-, 6-, or 7-dimensional models [39]. From theoretical
physics, the superstring theory proposes the space-time dimension could be 4
(Instain, Relativity), 5(Kalabi-Klein theory), 10 (Type 1 string), 11
(M-theory), or even 26 (Bosonic), etc. [33].

To deal with dimension varying and/or dimension uncertain systems, current
fixed dimension mathematics seems not very efficient. In recent years, we try
to use STP/STA to treat these kind of systems [15, 19, 20, 22]. Our previous
approaches are based on the dynamics over quotient space, which has nice
vector space structure. We then found that this approach is not suitable for
dealing with networked systems, which have higher dimensions and the
dimension-varying is relatively small. This fact stimulates us to investigate
the structure of real spaces with mixed dimensions itself. That is the real
motivation of the current paper.

By using the real space as the state-space of time-varying systems, it may
provide a practically useful frame for modeling and analyzing the time-varying
or uncertain systems and for designing controls.

Since the topic of this paper is wide and essential, what we did so far is
very limited and very elementary. Hence, this paper is only providing a
skeleton for what CDM should be. Further exploring and investigation are
necessary.

## References

  * [1] R. Abraham, J.E. Marden, Foundations of Mechanics, 2nd Ed., Benjamin/Cummings Pub., London, 1978.
  * [2] T. Akutsu, Algorithms for Analysis, Inference, and Control of Boolean networks, World Scientific, New Jersey, 2018.
  * [3] W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd Ed., Academic Press, Inc., Orlando, 1986.
  * [4] S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Springer, New York, 1981.
  * [5] D. Cheng, Semi-tensor product of matrices and its application to Morgen’s problem, Sci. China Ser. F, Inf. Sci., Vol. 44, No. 3, 2001.
  * [6] D. Cheng, H. Qi, Z. Li, Analysis and Control of Boolean Networks, A Semi-Tensor Product Approach, Springer, New York, 2011.
  * [7] D. Cheng, H. Qi, Y. Zhao, An Inroduction to Semi-Tensor Product of Matrices and Its Applications, World Scientific, Singapore, 2012.
  * [8] D. Cheng, H. Qi, F. He, Mappings and Dynamic Processes over Finite Sets - Semi-tensor Product Approach, Science Press, Beijing, 2016 (in Chinese).
  * [9] D. Cheng, H. Qi, Semi-tensor Product of Matrices - Theory and Application, Science Press, Beijing, 2007, 2-nd Ed., 2011, (in Chinese).
  * [10] D. Cheng, From Dimension-Free Matrix Theory to Cross-Dimensional Dynamic Systems, Elsevier, London, 1019.
  * [11] D. Cheng, On equivalence of Matrices, Asian J. Math., Vol. 23, No. 2, 257-348, 2019.
  * [12] D. Cheng, Z. Xu, T. Shen, Equivalence-based model of dimension-varying linear systems, IEEE Trans. Aut. Contr., Vol. 65, No. 12, 5444-5449, 2020.
  * [13] D. Cheng, H. Qi,  Lecture Notes in Semi-tensor Product of Matrices, Vol. 1,  Basic Theory and Multilinear Operation, Science Press, Beijing, 2020, (in Chinese).
  * [14] D. Cheng, Y. Wu, G. Zhao, S. Fu,  A comprehensive survey on STP approach to finite games,  J. Sys. Sci. Compl., Vol. 34, No. 5, 1666-1680, 2021.
  * [15] D. Cheng, Z. Xu, T. Shen, Equivalence-based model of dimension-varying linear systems, IEEE Trans. Aut. Contr., Vol. 65, No. 12, 5444-5449, 2020.
  * [16] D. Cheng, H. Qi,  Lecture Notes in Semi-tensor Product of Matrices, Vol. 1,  Basic Theory and Multilinear Operation, Science Press, Beijing, 2020, (in Chinese).
  * [17] D. Cheng, H. Qi,  Lecture Notes in Semi-tensor Product of Matrices, Vol. 2,  Analysis and Control of Logical Dynamic Systems, Science Press, Beijing, 2022, (in Chinese).
  * [18] D. Cheng, C. Li, Y. Hao, X. Zhang,  Lecture Notes in Semi-tensor Product of Matrices, Vol. 3,  Semi-tensor Product Approach to Finite Games, Science Press, Beijing, 2022, (in Chinese).
  * [19] D. Cheng, Z. Ji, From dimension-free manifolds to dimension-varying control systems, Commun. Inform. Sys., Vol. 23, No. 1, 85-150, 2023.
  * [20] D. Cheng, Z. Ji, Lecture Notes in Semi-tensor Product of Matrices, Vol. 4, Finite and Dimension-Free Dynamic Systems, Science Press, Beijing, 2023, (in Chinese).
  * [21] D. Cheng, J. Feng, J. Zhong, Y. Wu, K. Zhang, Lecture Notes in Semi-tensor Product of Matrices, Vol. 5, Application to Engineering and other Systems, Science Press, Beijing, 2024, (in Chinese).
  * [22] D. Cheng, From semi-tensor product of metrices to non-square general linear algebra and general linear group, (preprint: http:arxiv.org/abs/2305.19794v2, 2023), Commu. Inform. Sys., to appear.
  * [23] D. Cheng, X. Zhang, Z. Ji, C. Li, Observer-based realization of control systems, (preprint: http:arxiv.org/abs/2404.15688, 2024).
  * [24] G. Choquet, Topology, Academic Press, New York, 1966.
  * [25] J.B. Conway, A course in Functional Analysis, Springer-Verlag, New York, 1985.
  * [26] J.D. Dixon, B. Mortimer, Permtation Group, Springer-Verlag, New York, 1996,.
  * [27] J. Fang. In Introduction to Lattice Theory, High. Edu. Press, Beijing, 2014, (in Chinese).
  * [28] E. Fornasini, M.E. Valcher, Recent developments in Boolean networks control, J. Contr. Dec., Vol. 3, No. 1, 1-18, 2016.
  * [29] W. Greub, Linear Algebra, 4 ed., Springer-Verlag, New York, 1981.
  * [30] R. Huang, Z. Ye, An improved dimension-changeabel matrix model of simulationg the insect population dynamics, Entomological Knowledge, Vol. 32, No. 3, 162-164, 1995.
  * [31] T.W. Hungerford, Algebra, Springer-verlag, New York, 1974.
  * [32] P. Jiao, Y. Hao, J. Bin, Modeling and control of spacecraft formation based on impulsive switching with variable dimensions, Computer Simulation, Vol. 31, No. 6, 2014, 124-128.
  * [33] M. Kaku, Introduction to Supersting and M-Theory, 2nd Ed., Springer-Verlag, New York, 1999.
  * [34] H. Li, G. Zhao, M. Meng, J. Feng, A survey on applications of semi-tensor product method in engineering, Science China, Vol. 61, 010202:1-010202:17, 2018 .
  * [35] H. Li, G. Zhao, P. Guo, Z. Liu, Analysis and Control of Finite-Valued Systems, CRC Press, 2018.
  * [36] Z. Liu, H. Qiao, S-System Theory of Semigroup, 2nd Ed., Science Press, Beijing, 2008 (in Chinese).
  * [37] Y. Liu, J. Liu, L. Sun, Sampled-date Control of Logical Networks, High Education Press, Springer, Singapore, 2023.
  * [38] J. Lu, H. Li, Y. Liu, F. Li, Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Contr. Thm& Appl., Vol. 11, No. 13, 2040-2047, 2017.
  * [39] J. Machowski, J.W. Bialek, J.R. Bumby, Power System Dynamics and Stability, John Wiley and Sons, Inc., Chichester, 1997.
  * [40] S. Mei, F. Liu, A. Xie, Semi-tensor Product Approach for Transient Process Analysis of Power Systems, Tsinghua Univ. Press, Beijing, 2010, (in Chinese).
  * [41] A. Muhammad, A. Rushdi, F.A. M. Ghaleb, A tutorial exposition of semi-tensor products of matrices with a stress on their representation of Boolean function, JKAU Comp. Sci., Vol. 5, 3-30, 2016.
  * [42] , J.L. Kelley, General Topology, Springer-Verlag, New York, 1955.
  * [43] H.H. Schaefer, Topological Vector Spaces, Springer-Verlag, New York, 1970.
  * [44] A.E. Taylor, D.C. Lay, Introduction to Functional Analysis, John Wiley & Sons Inc., New York, 1980.
  * [45] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Springer-Verlag, New York, 1984.
  * [46] S.E. Boolean Systems - Topics in Asynchronicity, Elsevier, London, 2023.
  * [47] H. Wang, J. Wang, X. Bian, G. Fu, SLAM of AUV based on the combined EKF, Robot, Vol. 34, No. 1, 56-64, 2012.
  * [48] W.M. Wonham, Linear Multivariable Control, A Geometric Approach, Springer-Verlag, Berlin, 1974.
  * [49] R. Xu, L. Liu, Q. Zhu, J. Shen, Application of a dimension-changeabel matrix model on the simulation of the population dynamics of greenhouse whiteflies, ACTA Ecological Sinica, Vol. 1, No. 2, 147-158, 1981.
  * [50] H. Yang, B. Jiang, V. Cocquempot, Stabilization of Switched Nonlinear Systems with Unstable Modes, Chapter 4. Switched Nonlinear Systems with Varying states, Springer, Switzerland, 2014.
  * [51] Y. Yan, D. Cheng, J. Feng, H. Li, J. Yue, Survey on applications of algebraic state space theory of logical systems to finite state machines, Sci. China, Inform. Sci., //doi.org/10.1007/s11432-022-3538-4.
  * [52] K. Zhang, L. Zhang, L. Xie, Discrete-Space Dynamic Systems, Springer, New York, 2020.
  * [53] J. Zhong, D. Lin, On minimum period of nonlinear feedback shift registers in Grain-like structure, IEEE Trans. Inform. Theory, Vol. 64, No. 9, 6429-6442, 2018.