# Towards a Singular Value Decomposition and spectral theory for all rings

Ran Gutin111Department of Computer Science, Imperial College London

(27th August 2024)

###### Abstract

We propose definitions of SVD, spectral decomposition (for self-adjoint
matrices) and Jordan decomposition which make sense for all rings. For many
rings, these decompositions can be shown to exist. For some specific rings,
these decompositions are complicated to describe in full and prove the
existence of. These decompositions have occurred piecemeal in the literature.
We conjecture that they exist for many rings, including all Clifford algebras
over the real numbers and complex numbers. The origin of this programme is not
directly in module theory or linear algebra.

##  1 Motivation before giving the definitions

###  1.1 Objective

The objective of this paper is to propose a general definition for three
matrix decompositions, which can be shown to be satisfiable (or not) over any
given ring:

  * •

SVD (Singular Value Decomposition)

  * •

Spectral decomposition. In this paper, we see this as a decomposition of self-
adjoint matrices.

  * •

Jordan decomposition.

The motivation for this is that we have discovered analogues of those three
decompositions outside the obvious linear algebra or module theory context.
These analogues present canonical forms which are not as simple as in the
complex case, because they might fail to be diagonal. This phenomenon is
already familiar for the Jordan decomposition (in the complex case), but is
not present for the other two decompositions when working over the complex
numbers. In general, the canonical forms can be complicated, and their
existence is not easy to prove on a case-by-case basis.

###  1.2 Summary of origins

I give an account of how I stumbled upon SVDs over different rings.
Admittedly, this is somewhat subjective. It is written in the first person.

There is a 19th century “non-Euclidean” geometry (in the Kleinian sense)
called _Laguerre geometry_ which admits a relationship to the algebra of
$2\times 2$ matrices over the dual numbers. I found that based on the
correspondence between the _congruence transformations_ of this geometry, and
the $2\times 2$ matrices over the dual numbers, there was a strong hint
towards the existence of an analogue of the Singular Value Decomposition over
the ring of dual numbers instead of the usual ring of complex numbers. An
account of this matrix/transformation correspondence can be found in Yaglom’s
book _Complex numbers in geometry_. The SVD interpretation and subsequent
result was inspired by a classification of the Laguerre transformations which
can be found in Yaglom’s book. This analogue of the SVD turned out to exist
for matrices over the dual numbers, where the matrices could be of any
possible dimension (not just the $2\times 2$ case that Yaglom considered, and
not just the square matrices), even in the case where the matrices were
singular, and satisfied certain uniqueness properties similar to the complex
SVD.

I will now state exactly the result I obtained [1]. The ring of dual numbers
is denoted formally as $\mathbb{R}[\varepsilon]/(\varepsilon^{2})$. 222More
informally, a dual number is a number of the form $a+b\varepsilon$ where
$\varepsilon^{2}=0$ while $\varepsilon\neq 0$. The dual numbers form an
associative and commutative unital algebra over the real numbers. The dual
numbers admit an involution $*$ for which
$(a+b\varepsilon)^{*}=a-b\varepsilon$. This defines a conjugate-transpose or
adjoint operator on matrices $M^{*}$ such that $(M^{*})_{ij}=(M_{ji})^{*}$.
Analogously, we get a notion of unitary matrix $U$ for which
$UU^{*}=U^{*}U=I$. Given a matrix $M$ over the dual numbers (of any
dimension), I obtained a result which states that $M$ can be factorised into
$USV^{*}$ where $U$ and $V$ are unitary, and $S$ is a direct sum of matrices
in the set

$\displaystyle\\{[x]:x\in\mathbb{R};x>0\\}$
$\displaystyle\cup\left\\{\begin{bmatrix}x&-y\varepsilon\\\
y\varepsilon&x\end{bmatrix}:x\in\mathbb{R},y\in\mathbb{R};x>0,y>0\right\\}$
$\displaystyle\cup\\{[y\varepsilon]:y\in\mathbb{R};y>0\\}$
$\displaystyle\cup\\{0_{1,0},0_{0,1}\\}.$

This set is quite complicated, and we won’t dwell on it. Notice though that
the matrix $0_{1,0}$ (and $0_{0,1}$) is missing in the usual account of the
SVD over $\mathbb{C}$, but we will show that it is implicitly there as well!
It’s the unique $1\times 0$ matrix.

To explain the occurrence of this strange matrix, let’s consider the complex
matrix $M=\begin{bmatrix}1&2&0\\\ 2&1&0\end{bmatrix}$. Ignoring the exact
values of $U$ and $V$, but focussing our attention on $S$, we have that
$M=USV^{*}$ where $S=[3]\oplus[-1]\oplus 0_{0,1}$. The reader should verify
that given a matrix $K$, the value of $K\oplus 0_{0,1}$ is the same matrix as
$K$ but padded with a zero column. Likewise, the value of $K\oplus 0_{1,0}$ is
the same matrix as $K$ but padded with a zero row. The value of
$[3]\oplus[-1]\oplus 0_{0,1}$ is thus $\begin{bmatrix}3&0&0\\\
0&-1&0\end{bmatrix}$ as expected. We conclude that the “singular values” of a
complex matrix form a multiset (a similar object to a _set_ but the elements
are allowed to repeat) whose elements belong to the set
$\\{[x]:x\in\mathbb{R};x>0\\}\cup\\{0_{1,0},0_{0,1}\\}$. This is actually a
set of _matrices_ rather than scalar “values”. This suggests that the notion
of a singular value as a scalar value is questionable.

Given a matrix $M$ over $\mathbb{C}$ or
$\mathbb{R}[\varepsilon]/(\varepsilon^{2})$, the direct summands that make up
$S$ in $M=USV^{*}$ are unique up to permutation. In the dual number case, a
direct summand can be of dimension $2\times 2$, and non-square direct summands
can be found over both rings (and in fact all rings).

Upon discovering the above dual number SVD, I found that a dual number SVD of
sorts had already been considered in the literature, and it wasn’t the one
above. In that case, the involution over the dual numbers is the trivial one:
$z^{*}:=z$, and the notion of unitary matrix degenerates into
$U^{T}U=UU^{T}=I$. We then have that the direct summands that make up $S$ are
of the form:

$\\{[x+y\varepsilon]:x\in\mathbb{R},y\in\mathbb{R};x>0\\}\cup\\{[y\varepsilon]:y\in\mathbb{R};y>0\\}\cup\\{0_{1,0},0_{0,1}\\}.$

This shows that the correct setting is not a _ring_ as such, but a _$*$
-ring_, which is a ring equipped with an involution. The motivation for
considering matrix algebra over this particular $*$-ring came from the
mechanics literature where this SVD was suggested for solving kinematic
synthesis problems.

Over the $*$-ring of _double numbers_ , which are defined as
$\mathbb{R}\oplus\mathbb{R}$, equipped with the involution $(a,b)^{*}:=(b,a)$,
a matrix decomposition that constitutes an SVD over the double numbers has
been given [3]. I attempted to come up with such a decomposition myself, which
I called the Jordan SVD, but it had the problem that it didn’t exist for all
matrices over the double numbers, and therefore didn’t satisfy the general
criteria I give here. The paper in which the _actual_ double-number SVD is
introduced has some limitations, chief among them is that the paper is unaware
of the connection to the double numbers, or that the decomposition constitutes
_the_ analogue of the SVD over the $*$-ring of double numbers. A lot of
terminology is introduced there, like _contragredient equivalence_ , which I
view as needless.

Finally, over the _quaternions_ an analogue of SVD exists which satisfies my
definition. This has recently been extended to the case of the _dual
quaternions_ (an algebra equal to
$\mathbb{H}[\varepsilon]/(\varepsilon^{2})$). These algebras are non-
commutative, so our definitions don’t assume commutativity.

There is clearly a need for a general result which states precisely when an
analogue of the SVD, spectral decomposition for self-adjoint matrices, or
Jordan decomposition, exists for a given $*$-ring. Such a result has already
been obtained for the analogue of the Jordan decomposition which we discuss
here, but this appears at the moment to be the least general of the three
decompositions. We supply general definitions here and formulate conjectures.

###  1.3 Rough desired criteria

If $M=USV^{*}$ is the “SVD” of M, then $S$ should be a block diagonal matrix
which is unique up to permutation of the blocks. This is similar to, and
somewhat related to, the fact that the Jordan decomposition of a square matrix
$M$, expressed as $PJP^{-1}$, should be unique up to permutation of the Jordan
blocks. The objective is to decide whether or not a set of such blocks can be
exhibited for a given ring.

The formal definitions which we give below express in a rigorous way the
desired existence and uniqueness properties which we’ve alluded to.

##  2 Definitions

A $*$-ring is a ring $R$ equipped with an automorphism denoted $*:R\to R$ that
has order $2$. Such an automorphism is called an _involution_. Every ring can
be made into a $*$-ring in at least one way by defining $z^{*}=z$ for all
$z\in R$.

We define the following monoids:

  * •

$\operatorname{Mat}(R,*)$ is the monoid of all matrices over a $*$-ring $R$
where the monoid operation is $\oplus$, denoting direct sum of matrices. The
matrices can be of any possible dimensions, and they don’t have to be square.
The matrices can also have $0$ rows or $0$ columns.

  * •

$\operatorname{Herm}(R,*)$ is the monoid of all self-adjoint matrices over a
$*$-ring $R$ where the monoid operation is $\oplus$.

  * •

$\operatorname{Sq}(R)$ is the monoid of all square matrices over a ring $R$,
where $R$ is merely a ring and not a $*$-ring.

We then define the following equivalence relations on
$\operatorname{Mat}(R,*)$ and its various submonoids above:

  * •

${\sim_{\text{UE}}}$ means _unitary equivalence_. In other words, we have that
$A{\sim_{\text{UE}}}B$ is true whenever there exist unitary matrices $U$ and
$V$ such that $A=UBV^{*}$.

  * •

${\sim_{\text{US}}}$ means _unitary similarity_. In other words, we have that
$A{\sim_{\text{US}}}B$ is true whenever there exists a unitary matrix $V$ such
that $A=VBV^{*}$.

  * •

${\sim_{\text{S}}}$ means _similarity_. In other words, we have that
$A{\sim_{\text{S}}}B$ is true whenever there an invertible matrix $P$ such
that $A=PBP^{-1}$.

We aim to study the three monoids:

  * •

$\overline{\operatorname{Mat}}(R,*):=\operatorname{Mat}(R,*)/{\sim_{\text{UE}}}$
with the intention of generalising the _singular value decomposition_ ,

  * •

$\overline{\operatorname{Herm}}(R,*):=\operatorname{Herm}(R,*)/{\sim_{\text{US}}}$
with the intention of generalising the _spectral theorem_ (on self-adjoint
matrices).

  * •

$\overline{\operatorname{Sq}}(R,*):=\operatorname{Sq}(R)/{\sim_{\text{S}}}$
with the intention of generalising the _Jordan decomposition_.

To do this, notice that all three monoids are abelian, and for some $*$-rings
$R$, they are even _free_ as abelian monoids. A _free abelian monoid_ consists
of all finite multisets whose elements belong to some set $S$. An isomorphism
between each of the three monoids above and a free abelian monoid produces an
analogue of the SVD, spectral theorem, and Jordan decompositions respectively.

As an aside: Note that all six parametrised families of monoids can be thought
of as _functors_ between two categories, if this fact can ever be useful. The
functors are all from the category of $*$-rings to the category of monoids.

##  3 Discussion of the relationship of monoid algebra to the SVD, spectral
decomposition and Jordan decompositions

When an abelian monoid $(M,+,0)$ is free, there is a _unique_ subset (which we
will here denote $P$) of $M$ (called the generators of $M$) such that each
element of $M$ is a unique sum elements of the generators $P$. It can be
argued that many “unique factorisation” type results in mathematics merely
state the fact that some abelian monoid with a complicated construction is
actually free.

An example of the above is the abelian monoid of _positive integers_ under
integer multiplication, in which the generators are the prime numbers. The
fact that for every integer _there exists_ a prime factorisation, and moreover
it is _unique_ , is equivalent to the fact that the abelian monoid of positive
integers is free.

The abelian monoid $\overline{\operatorname{Mat}}(R,*)$ for a given $*$-ring
$(R,*)$ is constructed in a complicated way. In spite of that fact, if it is
free, then it is actually quite simple. Its freeness captures the existence
and uniqueness of the SVD.

##  4 Summary of existing results

Theorem 1: $\overline{\operatorname{Sq}}(R)$ is a free abelian monoid for
every Artinian ring $R$.

_Proof_. See [4].

The above theorem in particular shows that something approaching a Jordan
decomposition exists for the dual numbers, which are an Artinian ring. All
finite dimensional associative algebras over a field, like the quaternions and
dual numbers, are Artinian, so this generalises an important aspect of the
Jordan decomposition.

Let $\operatorname{swap}(a,b):=(b,a)$. This is a natural choice of involution
for $R\oplus R$ where $R$ is any ring. An illustrative special case is when
$R$ is $\mathbb{R}$, in which case $R\oplus R$ is commonly called either the
_split-complex numbers_ or the _double numbers_. This is an interesting
hypercomplex number system considered in [2]. When $R=\mathbb{C}$, the name we
give to $R\oplus R$ is the _double complex numbers_.

Theorem 2: $\overline{\operatorname{Herm}}(R\oplus
R,\operatorname{swap})\cong\overline{\operatorname{Sq}}(R)$ for every ring
$R$.

Remark: When both sides are treated as functors of $R$, the isomorphism is
natural. We don’t attempt to verify this fact.

_Proof_. Let $M$ be an element of $\operatorname{Herm}(R\oplus
R,\operatorname{swap})$. We have that $M=(A,B^{T})$ because $M$ in particular
must belong to $J(R\oplus R)$. We furthermore have that $M$ is Hermitian, so
$(A,B^{T})=M=M^{*}=(B,A^{T})$; therefore $A=B$. In summary, we have that
$M=(A,A^{T})$. Let
$\phi_{R}:\operatorname{Sq}(R)\,\to\operatorname{Herm}(R\oplus
R,\operatorname{swap})$ be given by $\phi_{R}(A)=(A,A^{T})$. The mapping is
clearly an isomorphism of monoids. But we’re not done yet because we would
like an isomorphism
$\psi_{R}:\overline{\operatorname{Sq}}(R)\to\overline{\operatorname{Herm}}(R\oplus
R,\operatorname{swap})$ instead. This isomorphism is obtained by noticing that
if $M\sim_{\text{S}}K$ then $\phi_{R}(M)\sim_{\text{US}}\phi_{R}(K)$. We show
that this is indeed true: If $M{\sim_{\text{S}}}K$ then $M=PKP^{-1}$, so
$\phi_{R}(M)=(M,M^{T})=(PKP^{-1},(P^{T})^{-1}K^{T}P^{T})=(P,(P^{-1})^{T})(K,K^{T})(P^{-1},P^{T}){\sim_{\text{US}}}(K,K^{T})=\phi_{R}(K)$.
We are done.

The above result is significant because it shows that the monoid family
$\overline{\operatorname{Sq}}(R)$ is somewhat redundant. The study of these
monoids is subsumed by the study of monoids in the family
$\overline{\operatorname{Herm}}(R,*)$.

Theorem 3: $\overline{\operatorname{Mat}}(R,*)$ admits an isomorphism to a
free abelian monoid with the following generators when $(R,*)$ is any of the
following $*$-rings:

| $R$ | $*:R\to R$ | Generators of $\overline{\operatorname{Mat}}(R,*)$
---|---|---|---
1 | $0$ | $z^{*}:=z$ | $\\{0_{1,0},0_{0,1}\\}$
2 | $\mathbb{R}$ | $z^{*}:=z$ | $\begin{aligned} &\\{[x]:x\in\mathbb{R};x>0\\}\\\ &\cup\\{0_{1,0},0_{0,1}\\}\end{aligned}$
3 | $\mathbb{C}$ | $(a+bi)^{*}:=a-bi$ | $\begin{aligned} &\\{[x]:x\in\mathbb{R};x>0\\}\\\ &\cup\\{0_{1,0},0_{0,1}\\}\end{aligned}$
4 | $\mathbb{R}[\varepsilon]/(\varepsilon^{2})$ | $z^{*}:=z$ | $\begin{aligned} &\\{[x+y\varepsilon]:x\in\mathbb{R},y\in\mathbb{R};x>0\\}\\\ &\cup\\{[y\varepsilon]:y\in\mathbb{R};y>0\\}\cup\\{0_{1,0},0_{0,1}\\}\end{aligned}$
5 | $\mathbb{R}[\varepsilon]/(\varepsilon^{2})$ | $(a+b\varepsilon)^{*}:=a-b\varepsilon$ | $\begin{aligned} &\\{[x]:x\in\mathbb{R};x>0\\}\\\ &\cup\left\\{\begin{bmatrix}x&-y\varepsilon\\\ y\varepsilon&x\end{bmatrix}:x\in\mathbb{R},y\in\mathbb{R};x>0,y>0\right\\}\\\ &\cup\\{[y\varepsilon]:y\in\mathbb{R};y>0\\}\cup\\{0_{1,0},0_{0,1}\\}\end{aligned}$
6 | $\mathbb{C}\oplus\mathbb{C}$ | $\operatorname{swap}$ | $\begin{aligned} &\\{(J_{m}(re^{i\theta}),J_{m}(re^{i\theta})^{T}):r>0,\theta\in[0,\pi)\\}\\\ &\cup\left\\{\left(\begin{pmatrix}I_{m}&0\end{pmatrix},\begin{pmatrix}0\\\ I_{m}\end{pmatrix}^{T}\right):m\in\mathbb{N}\right\\}\\\ &\cup\left\\{\left(\begin{pmatrix}0\\\ I_{m}\end{pmatrix},\begin{pmatrix}I_{m}&0\end{pmatrix}^{T}\right):m\in\mathbb{N}\right\\}\\\ &\cup\\{(J_{m}(0),I_{m}^{T}):m\in\mathbb{N}\\}\\\ &\cup\\{(I_{m},J_{m}(0)^{T}):m\in\mathbb{N}\\}\cup\\{0_{1,0},0_{0,1}\\}\end{aligned}$
7 | $\mathbb{H}$ | $\begin{aligned} &(a+bi+cj+dk)^{*}:=\\\ &a-bi-cj-dk\end{aligned}$ | $\\{[x]:x\in\mathbb{R};x>0\\}\cup\\{0_{1,0},0_{0,1}\\}$
8 | $\begin{aligned} &\operatorname{span}(1,i,\varepsilon j,\varepsilon k)\\\ &\subset\mathbb{H}[\varepsilon]/(\varepsilon^{2})\end{aligned}$ | $\begin{aligned} &(a+bi+c\varepsilon j+d\varepsilon k)^{*}:=\\\ &a-bi-c\varepsilon j-d\varepsilon k\end{aligned}$ | $\begin{aligned} &\\{[x]:x\in\mathbb{R};x>0\\}\\\ &\cup\left\\{\begin{bmatrix}x&-\delta\\\ \delta&x\end{bmatrix}:x\in\mathbb{R};x>0,\delta^{2}=0\right\\}\\\ &\cup\\{[y\varepsilon j]:y\in\mathbb{R};y>0\\}\cup\\{0_{1,0},0_{0,1}\\}\end{aligned}$

_Proof_. For each example in turn:

  1. 1.

The ring here is the $0$ ring, whose set of elements is $\\{0\\}$. The
definition of the operations $\\{+,-,\times\\}$ is immediate. The generators
are the two degenerate matrices $0_{1,0}$ and $0_{0,1}$. The first matrix has
$1$ row and $0$ columns. The second matrix has $0$ rows and $1$ column.
Matrices with $0$ rows or columns appear strange but they are an inevitable
part of matrix algebra because they represent linear maps which map to or from
$0$-dimensional vector spaces.

  2. 2.

A proof of this fact can be found in any undergraduate textbook on linear
algebra for mathematics students, as long as that textbook covers the spectral
theorem and SVD. The only unusual generators are $0_{1,0}$ and $0_{0,1}$,
which are $1\times 0$ and $0\times 1$ matrices respectively. While it is
strange to have matrices with these dimensions, they are an inevitable part of
the matrix formalism because they represent linear maps that go to or from
$0$-dimensional vector spaces.

  3. 3.

A proof of this fact can be found in any undergraduate textbook on linear
algebra for mathematics students, as long as that textbook covers the spectral
theorem and SVD. The only unusual generators are $0_{1,0}$ and $0_{0,1}$,
which are $1\times 0$ and $0\times 1$ matrices respectively. While it is
strange to have matrices with these dimensions, they are an inevitable part of
the matrix formalism because they represent linear maps that go to or from
$0$-dimensional vector spaces.

  4. 4.

See [1]. The only thing missing from the paper is a complete proof of
uniqueness. This can be achieved by using the uniqueness of the
eigendecomposition for dual-number matrices (a fact proved in the reference)
and observing that the diagonal form of the block matrix
$\begin{pmatrix}0&M^{*}\\\ M&0\end{pmatrix}$ is $\Sigma\oplus(-\Sigma)$ where
$\Sigma$ is the normal form of $M$ under SVD. This implies that $\Sigma$ is
unique.

  5. 5.

See [1]. Uniqueness can be reduced to the case of the involution $z^{*}:=z$.

  6. 6.

This follows from [3].

  7. 7.

This follows from Theorem 7.2 of [7].

  8. 8.

See [5].

Theorem 4: $\overline{\operatorname{Herm}}(R,*)$ admits an isomorphism to a
free abelian monoid with the following generators when $(R,*)$ is any of the
following $*$-rings:

| $R$ | $*:R\to R$ | Generators of $\overline{\operatorname{Herm}}(R,*)$
---|---|---|---
1 | $0$ | $z^{*}:=z$ | $\\{[0]\\}$
2 | $\mathbb{R}$ | $z^{*}:=z$ | $\\{[x]:x\in\mathbb{R};x>0\\}\cup\\{[0]\\}$
3 | $\mathbb{C}$ | $(a+bi)^{*}:=a-bi$ | $\\{[x]:x\in\mathbb{R};x>0\\}\cup\\{[0]\\}$
4 | $\mathbb{R}[\varepsilon]/(\varepsilon^{2})$ | $z^{*}:=z$ | $\begin{aligned} &\\{[x+y\varepsilon]:x\in\mathbb{R},y\in\mathbb{R};x>0\\}\\\ &\cup\\{[y\varepsilon]:y\in\mathbb{R};y>0\\}\\\ &\cup\\{[0]\\}\end{aligned}$
5 | $\mathbb{R}[\varepsilon]/(\varepsilon^{2})$ | $(a+b\varepsilon)^{*}:=a-b\varepsilon$ | $\begin{aligned} &\\{[x]:x\in\mathbb{R};x>0\\}\\\ &\cup\left\\{\begin{bmatrix}x&-y\varepsilon\\\ y\varepsilon&x\end{bmatrix}:x\in\mathbb{R},y\in\mathbb{R};x>0,y>0\right\\}\\\ &\cup\\{y\varepsilon:y\in\mathbb{R};y>0\\}\\\ &\cup\\{[0]\\}\end{aligned}$
6 | $\mathbb{C}\oplus\mathbb{C}$ | $\operatorname{swap}$ | $\\{(J_{m}(re^{i\theta}),J_{m}(re^{i\theta})^{T}):r\geq 0,\theta\in[0,\pi)\\}$
7 | $\mathbb{H}$ | $\begin{aligned} &(a+bi+cj+dk)^{*}:=\\\ &a-bi-cj-dk\end{aligned}$ | $\\{[x]:x\in\mathbb{R};x>0\\}\cup\\{[0]\\}$
8 | $\begin{aligned} &\operatorname{span}(1,i,\varepsilon j,\varepsilon k)\\\ &\subset\mathbb{H}[\varepsilon]/(\varepsilon^{2})\end{aligned}$ | $\begin{aligned} &(a+bi+c\varepsilon j+d\varepsilon k)^{*}:=\\\ &a-bi-c\varepsilon j-d\varepsilon k\end{aligned}$ | $\begin{aligned} &\\{[x]:x\in\mathbb{R};x>0\\}\\\ &\cup\left\\{\begin{bmatrix}x&-\delta\\\ \delta&x\end{bmatrix}:x\in\mathbb{R};x>0,\delta^{2}=0\right\\}\\\ &\cup\\{[0]\\}\end{aligned}$
9 | $\mathbb{Z}$ | $z^{*}:=z$ |  | The generators are related to the set of
---
adjacency matrices of all connected graphs.
We don’t give an explicit description.
10 | $\mathbb{C}$ | $z^{*}:=z$ | Very complicated. Given in [6].

_Proof_. For each example in turn:

  1. 1.

The ring here is the $0$ ring, whose set of elements is $\\{0\\}$. The
definition of the operations $\\{+,-,\times\\}$ is immediate.

  2. 2.

A proof of this fact can be found in any undergraduate textbook on linear
algebra for mathematics students, as long as that textbook covers the spectral
theorem and SVD.

  3. 3.

A proof of this fact can be found in any undergraduate textbook on linear
algebra for mathematics students, as long as that textbook covers the spectral
theorem and SVD.

  4. 4.

See [1].

  5. 5.

See [1].

  6. 6.

Consequence of Theorem 2.

  7. 7.

This follows from Theorem 7.2 of [7].

  8. 8.

See [5].

  9. 9.

In this case, we haven’t stated the result clearly enough that we can prove
it. If we wished to, we _could_ state it rigorously and prove it. Note that a
unitary matrix $U$ that is also an integer matrix is precisely a signed
permutation matrix. Two undirected graphs, represented as adjacency matrix $M$
and $K$, are isomorphic iff there exists a permutation matrix $P$ such that
$M=PKP^{-1}$. The generators are therefore connected graphs which are combined
by direct sum to form unconnected graphs.

  10. 10.

See [6].

##  5 Meta-conjectures

By a meta-conjecture, we mean a conjecture which is true for a large class of
rings, but not necessarily for all rings. We suspect that the class of rings
for which these conjectures are true is large.

Conjecture 1: For any ring $*$-ring $(R,*)$,
$\overline{\operatorname{Herm}}(R,*)$ is free.

Conjecture 2: For any ring $*$-ring $(R,*)$,
$\overline{\operatorname{Mat}}(R,*)$ is free.

We know from theorem 1 that $\overline{\operatorname{Sq}}(R)$ is free whenever
$R$ is Artinian. $\overline{\operatorname{Sq}}(R)$ may be free for some non-
Artinian rings as well. Therefore we do not pose this as one of the meta-
conjectures.

A proof of those two meta-conjectures may end up being non-constructive, in
the sense that it might fail to give the generators of the corresponding free
abelian monoids explicitly. We consider the search for a constructive proof in
some special cases to be worthwhile.

## References

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