# A conjecture concerning $\ast$-algebras that unifies some matrix
decompositions

Ran Gutin
Imperial College London

###### Abstract

In this note, we propose a simple-looking but broad conjecture about star-
algebras over the field of real numbers. The conjecture enables many matrix
decompositions to be represented by star-algebras and star-ideals. This paper
is written for people with a background in representation theory and module
theory. The motivation for investigating this is the possibility of expressing
polymorphic algorithms in numerical and theoretical linear algebra. This is
similar to but different from algebraic (semiring based) approaches to dynamic
programming. We prove certain cases of the conjecture.

##  1 Introduction

In this note, we propose a conjecture about finite-dimensional $\ast$-algebras
(also called involution algebras) [17] which we think has applications in
numerical and theoretical linear algebra. Our conjecture is that a simple-
looking but broad generalisation of the spectral theorem is true for all
finite-dimensional $\ast$-algebras over the field $\mathbb{R}$. We provide a
proof for some cases.

We state the conjecture:

###### Conjecture 1

Let $H$ be a self-adjoint element (that is, satisfying $H^{\ast}=H$) of a
finite-dimensional $\ast$-algebra $\mathcal{A}$ over $\mathbb{R}$. Let
$k\in\mathbb{N}$ be the maximum number for which the following decomposition
exists:

$H=\sum_{i=1}^{k}P_{i}HP_{i},$

where for all $i,j\in[k]$:

  * $\bullet$

$P_{i}\in\mathcal{A},$

  * $\bullet$

$P_{i}^{\ast}=P_{i},$

  * $\bullet$

$P_{i}P_{j}=\delta_{ij}P_{i},$

  * $\bullet$

$P_{i}\neq 0$.

Consider another such decomposition for the same $H$:

$H=\sum_{i=1}^{k}Q_{i}HQ_{i}.$

Then there exists a permutation $\sigma\in S_{k}$ and a $U\in\mathcal{A}$ such
that:

  * $\bullet$

$U^{\ast}=U^{-1}$,

  * $\bullet$

$UHU^{\ast}=H$,

  * $\bullet$

for all $i\in[k]$: $UQ_{i}U^{\ast}=P_{\sigma(i)}.$

By assuming the conjecture, we can pick $\ast$-algebras which in some sense
represent certain matrix decompositions (with prior investigation here [9]).
By a matrix decomposition, we mean a way of writing matrices as a product of
other matrices – a connection which is partly justified by lemma 1. This
unification of some matrix decompositions emphasises the uniqueness aspect, as
opposed to the existence aspect (which is made trivial), of those
decompositions. One of the motivations is that the use of $\ast$-algebras
allows computer code written to compute one decomposition to be directly used
to compute different ones (related to but different from the somewhat well-
known ideas in [19, 13]). Note that we will not develop these computing
applications here because this is a theoretical paper.

We demonstrate the correspondence this creates using some examples below.

##  2 Consequences for different $\ast$-algebras

We’re going to show how this conjecture easily re-derives some existing matrix
decompositions. These re-derivations are justified by the observation (later
proven in lemma 1) that the conjecture is sometimes equivalent to the
statement that over certain $\ast$-algebras, the conjecture is equivalent to
the cancellation property of unitary similarity.

###  2.1 Matrices over the complex numbers, with conjugate-transpose as their
involution

Let $H$ be an $n\times n$ Hermitian matrix over $\mathbb{C}$. We recall the
spectral theorem from linear algebra:

$H=VDV^{\ast},$

and recall that we’re interested in decompositions of the form

$H=\sum_{i=1}^{k}P_{i}HP_{i},$

for maximum $k$. In fact, we may obtain this by taking each column of $V$,
which we will call $v_{i}$, and letting $P_{i}=v_{i}v_{i}^{\ast}$. We get
$k=n$ and that $P_{i}HP_{i}=\lambda_{i}v_{i}v_{i}^{\ast}$. So we have:

$H=\sum_{i=1}^{k}P_{i}HP_{i}=\sum_{i=1}^{n}\lambda_{i}v_{i}v_{i}^{\ast}.$

This illustrates (and sketches the proof for) the conjecture for $\mathbb{C}$,
and shows that the conjecture generalises the spectral theorem.

###  2.2 Matrices over the double numbers, with conjugate-transpose as their
involution

Let ${}^{2}\mathbb{R}$ denote the $*$-algebra of _double numbers_ : The
underlying algebra is $\mathbb{R}\oplus\mathbb{R}$, and the involution is
$(a,b)\mapsto(b,a)$.

Given any finite-dimensional (without star) $\mathbb{R}$-algebra $R$, we can
make a $\ast$-algebra $R\oplus R^{\operatorname{op}}$, with involution being
$(x,y)\mapsto(y,x)$ [10]. Note that for some such algebras $R$ (but not all),
there exists some (any) involution, and we therefore have
$R^{\operatorname{op}}\cong R$. In those cases, we have $R\oplus
R^{\operatorname{op}}\cong R\oplus R\cong R\otimes{}^{2}\mathbb{R}$, where the
algebra $R$ in the last expression is equipped with any involution.

Observe that the conjecture for $M_{n}(R)\oplus M_{n}(R)^{\operatorname{op}}$
is equivalent to the invariant subspace decomposition for matrices in
$M_{n}(R)$. The existence of such a decomposition is a corollary of the Krull-
Schmidt theorem [14] – albeit for $R\in\\{\mathbb{R},\mathbb{C},\mathbb{H}\\}$
it is better described as the Jordan Normal Form or primary decomposition.

In particular, this means that the conjecture for $M_{n}({}^{2}\mathbb{R})$ –
that is, $n\times n$ matrices over the double numbers – is equivalent to the
Jordan Normal Form for $\mathbb{R}$-matrices.

###  2.3 General picture

In general, we obtain a correspondence between on the one hand:

  * $\bullet$

Matrix decompositions,

and

  * $\bullet$

finite-dimensional $\ast$-algebras over $\mathbb{R}$ along with an ideal.

The corresponding decomposition is obtained by considering Hermitian matrices
over the given $*$-algebra whose elements all belong to the specified ideal,
and then (nearly all the time) considering block-diagonal canonical forms of
those matrices under unitary similarity.

We make this explicit using a table, where we exhaust all indecomposable 2 and
3-dimensional $\ast$-algebras (which can be done with the aid of the
Wedderburn-Malcev theorem) and consider the decomposition which the conjecture
implies for all of its matrix algebras:

Algebra |  Involution |  $\ast$-ideal |  Corresponding decomposition |  Tame?
---|---|---|---|---
$\mathbb{R}$ |  $\operatorname{id}$ |  $\langle 1\rangle$ |  Spectral theorem |  Y
$\mathbb{C}$ |  $a+bi\mapsto a-bi$ |  $\langle 1\rangle$ |  Spectral theorem for $\mathbb{C}$-Hermitian matrices |  Y
$\mathbb{C}$ |  $\operatorname{id}$ |  $\langle 1\rangle$ |  Complex-symmetric spectral theorem |  Y111follows from observation that all square $\mathbb{C}$-matrices are similar to $\mathbb{C}$-symmetric matrices, combined with the polar decomposition trick given for instance in [15]
$\mathbb{R}[X]/(X^{2})$ |  $\operatorname{id}$ |  $\langle 1\rangle$ |  Spectral decomposition of $H+\varepsilon H^{\prime}+o(\varepsilon)$ where $H,H^{\prime}\in M_{n}(\mathbb{R})$ and both are symmetric |  Y [4, 8]
$\mathbb{R}[X]/(X^{2})$ |  $a+bX\mapsto a-bX$ |  $\langle 1\rangle$ |  Spectral decomposition of $H+\varepsilon H^{\prime}+o(\varepsilon)$ where $H,H^{\prime}\in M_{n}(\mathbb{R})$; $H$ is symmetric; $H^{\prime}$ is skew-symmetric |  Y [4, 8]
$\mathbb{R}[X]/(X^{2})$ |  $a+bX\mapsto a-bX$ |  $\langle X\rangle$ |  Spectral theorem for skew-symmetric $\mathbb{R}$-matrices |  Y
$\mathbb{R}\oplus\mathbb{R}$ |  $(a,b)\mapsto(b,a)$ |  $\langle 1\rangle$ |  Jordan Normal Form |  Y
$(\mathbb{R}\oplus\mathbb{R})+\mathbb{R}\delta$ 222with relations $\delta^{2}=0$, $\delta(x,y)=(y,x)\delta=x\delta$ |  Unique one where $\delta\mapsto\delta$ |  $\langle 1\rangle$ |  Canonical basis for a pair $(Q,L)$ consisting of a quadratic form $Q$ and an operator $L$ self-adjoint with respect to $Q$ |  Y [5]
$(\mathbb{R}\oplus\mathbb{R})+\mathbb{R}\delta$ LABEL:threed-arithmetic |  Unique one where $\delta\mapsto-\delta$ |  $\langle 1\rangle$ |  Canonical basis for a pair $(\omega,L)$ consisting a of symplectic form $\omega$ and an operator $L$ self-adjoint with respect to $\omega$ |  Y [5]
$(\mathbb{R}\oplus\mathbb{R})+\mathbb{R}\delta$ LABEL:threed-arithmetic |  Unique one where $\delta\mapsto\delta$ |  $\langle\delta\rangle$ |  Sylvester’s Law of Inertia |  Y
$(\mathbb{R}\oplus\mathbb{R})+\mathbb{R}\delta$ LABEL:threed-arithmetic |  Unique one where $\delta\mapsto-\delta$ |  $\langle\delta\rangle$ |  Analogue of Sylvester’s Law of Inertia for skew-symmetric matrices333The decomposition is for when $M=-M^{T}\in M_{n}(\mathbb{R})$; then there is an invertible $P\in M_{n}(\mathbb{R})$ such that $PMP^{T}=\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}^{\oplus\operatorname{rank}(M)/2}\oplus[0]^{\oplus\operatorname{null}(M)}$ |  Y
$\mathbb{R}[X]/(X^{3})$ |  $\operatorname{id}$ |  $\langle 1\rangle$ |  2nd-order perturbation theory of spectral theorem |  Y [4]
$\mathbb{R}[X]/(X^{3})$ |  $X\mapsto-X$ |  $\langle 1\rangle$ |  2nd-order perturbation theory of invariant subspace decomposition where skew-symmetric part is infinitesimal |  Y [4]
$\mathbb{R}[X]/(X^{3})$ |  $X\mapsto-X$ |  $\langle X\rangle$ |  Spectral decomposition of $H+\varepsilon H^{\prime}+o(\varepsilon)$ where $H,H^{\prime}\in M_{n}(\mathbb{R})$; $H$ is skew-symmetric; $H^{\prime}$ is symmetric |  Y
$\mathbb{R}[X,Y]/(X^{2},Y^{2},XY)$ |  $X\mapsto X,Y\mapsto-Y$ |  $\langle 1\rangle$ |  Spectral decomposition of $H+\varepsilon H^{\prime}+o(\varepsilon)$ where $H,H^{\prime}\in M_{n}(\mathbb{R})$; $H$ is symmetric; $H^{\prime}$ is an _arbitrary_ matrix |  N444Wild even when restricted to an ideal
$\mathbb{R}[X,Y]/(X^{2},Y^{2},XY)$ |  $X\mapsto X,Y\mapsto Y$ |  $\langle 1\rangle$ |  1st-order perturbation theory of spectral theorem, with 2 independent perturbations in symmetric directions |  NLABEL:wildeven
$\mathbb{R}[X,Y]/(X^{2},Y^{2},XY)$ |  $X\mapsto-X,Y\mapsto-Y$ |  $\langle 1\rangle$ |  1st-order perturbation theory of spectral theorem, with 2 independent perturbations in skew-symmetric directions |  NLABEL:wildeven
$\mathbb{R}[X,Y]/(X^{2},Y^{2},XY)$ |  $X\mapsto X,Y\mapsto-Y$ |  $\langle X,Y\rangle$ |  Block-diagonal form for $\mathbb{R}$-matrices under orthogonal similarity |  N [18, sec. 4]
$\mathbb{R}[X,Y]/(X^{2},Y^{2},XY)$ |  $X\mapsto X,Y\mapsto Y$ |  $\langle X,Y\rangle$ |  Block-diagonal form for pairs of symmetric $\mathbb{R}$-matrices under orthogonal similarity. |  N [18, sec. 4]
$\mathbb{R}[X,Y]/(X^{2},Y^{2},XY)$ |  $X\mapsto-X,Y\mapsto-Y$ |  $\langle X,Y\rangle$ |  Block-diagonal form for pairs of skew-symmetric $\mathbb{R}$-matrices under orthogonal similarity |  N [18, sec. 4]

The above lists all 2- and 3-dimensional cases. Note that we treat much of the
perturbation theory of matrix decompositions as – in some sense – being matrix
decompositions in their own right.

We also consider some notable 4-dimensional cases. There are too many cases to
exhaustively list here, so we’ve listed only a few below. Pay attention to the
corresponding decompositions. Note that these are not the only tame ones in 4
dimensions.

Algebra |  Involution |  $\ast$-ideal |  Corresponding decomposition |  Tame?
---|---|---|---|---
$(a,b)+\delta(a^{\prime},b^{\prime})$ 555where $\delta(a^{\prime},b^{\prime})=(b^{\prime},a^{\prime})\delta$ and $\delta^{2}=0$ |  Any which sends $(x,y)\mapsto(y,x)$ |  $\langle\delta\rangle$ |  Singular Value Decomposition |  Y
$(a+bi)+\delta(a^{\prime}+b^{\prime}i)$ 666where $\delta i=-i\delta$ and $\delta^{2}=0$ |  $\delta\mapsto\delta,i\mapsto-i$ |  $\langle\delta\rangle$ |  Autone-Takagi decomposition [11] |  Y
$(a+bi)+\delta(a^{\prime}+b^{\prime}i)$ LABEL:where-takagi-alg |  $\delta\mapsto-\delta,i\mapsto-i$ |  $\langle\delta\rangle$ |  Skew-symmetric Takagi decomposition |  Y
$M_{2}(\mathbb{R})$ |  Matrix adjugate |  $\langle 1\rangle$ |  “Symplectic spectral theorem”: Analogue of the spectral theorem for $2n\times 2n$ symplectic-self adjoint matrices under symplectic similarity |  Y777Tame for similar reasons to case $(\mathbb{C},\operatorname{id},\langle 1\rangle)$
$\mathbb{R}[X,Y]/(X^{2},Y^{2})$ |  $X\mapsto-X,Y\mapsto Y$ |  $\langle X\rangle$ |  Spectral decomposition of $H+\varepsilon H^{\prime}+o(\varepsilon)$ where $H$ and $H^{\prime}$ are skew-symmetric |  Y

By taking tensor products of $\ast$-algebras and their ideals, one can combine
features of different decompositions. For instance:

  * •

The somewhat intricate – but still tame! – conjugate-less analogue of SVD for
complex matrices considered in [12] is obtained by tensoring the
representative of the SVD – which is 4-dimensional – with the representative
for the complex-symmetric spectral theorem – which is 2-dimensional. It should
be intuitively plausible that this combines their features in a compelling
way.

  * •

Referring to the same paper [12], the main decomposition there can be seen as
the analogue of Singular Value Decomposition for matrices over the $*$-algebra
of double numbers888Recall that this is the algebra
$\mathbb{R}\oplus\mathbb{R}$ equipped with the involution $(a,b)\mapsto(b,a)$.
Since a matrix over the double numbers is essentially a pair of real matrices,
this creates an analogue of SVD for pairs of real matrices. Equivalently, this
may be obtained by using tensor products: A representative of the main
decomposition of that paper can be obtained by tensoring the triple
representing the Singular Value Decomposition with the one respresenting the
Jordan Normal Form. The resulting triple gives an involution and an ideal of
an $8$-dimensional algebra.

##  3 Obstacles to a general proof

While there isn’t much literature on $\ast$-algebras, there is plenty on
$C^{\ast}$-algebras. Unfortunately, much of the machinery does not generalise
well to $\ast$-algebras, as we will now show [17]. And already in 2
dimensions, most $\ast$-algebras are not $C^{\ast}$-algebras. This situation
is particularly interesting because much of the general theory in the
literature concerning $\ast$-algebras that aren’t $C^{\ast}$-algebras appears
to concern algebras consisting of unbounded (that is, discontinuous) linear
operators on infinite-dimensional Hilbert spaces – it’s difficult to apply
their methods to problems like this one.

One piece of machinery in the literature is “star representations” or
$\ast$-representations. This is the same thing as a representation of an
algebra, but with the additional requirement that the involution $\ast$ be
sent to the conjugate-transpose on complex matrices. This is already
impossible for most 2-dimensional $\ast$-algebras. We therefore have a curious
situation that while every algebra has a representation, almost no
$\ast$-algebra has a $\ast$-representation.

Much of the representation theory of topological groups is developed for
compact groups [1]. However, the unitary elements of our $\ast$-algebras
usually don’t form compact groups.

We’ve noticed that results in the literature which claim to solve
(effectively) special cases of our conjecture often have erroneous proofs. For
instance, we’ve tried to find generalisations of Specht’s theorem [15, 11] to
prove additional cases of our conjecture, but the proofs we found in the
literature for those generalisations were wrong. We confirmed this with the
authors. We will not cite examples for obvious reasons.

Trying to generalise the conjecture may be fraught: The conjecture over
arbitrary fields is false, as follows from basic Witt theory and the failure
of Sylvester’s Law of Inertia (undermining a prediction of the generalised
conjecture) over such fields. In spite of these obstacles, some steps to
produce something like a theory of matrix decompositions for finite fields
have been done [7]. There is prior work unifying the classical groups [10]
instead of the decompositions (with an eye towards algebraic K-theory), but
the structures considered (form rings, form modules) are quite different, and
we are not sure how to apply those general tools here.

##  4 Proofs of certain cases

We can prove some cases of the conjecture.

###### Definition 1

Self-adjoint matrices over a $\ast$-algebra $\mathcal{A}$ satisfy the
cancellation property under unitary similarity if whenever $A$ is unitarily
similar to $A^{\prime}$ and $A\oplus B$ is unitarily similar to
$A^{\prime}\oplus B^{\prime}$, then $B$ is unitarily similar to $B^{\prime}$.
Note that the matrices $A,A^{\prime},B,B^{\prime}$ are understood to be self-
adjoint.

###### Lemma 1

If for a given local $\ast$-algebra $\mathcal{A}$, if the self-adjoint
matrices over $\mathcal{A}$ satisfy the cancellation property under unitary
similarity, then for all $n\in\mathbb{N}$ the conjecture holds for
$M_{n}(\mathcal{A})$.

Proof  Consider $H=\sum_{i=1}^{k}P_{i}HP_{i}=\sum_{i=1}^{k}Q_{i}HQ_{i}$ for
largest $k$.

Use $\operatorname{im}(P_{i})$ for each $i$. Clearly,
$\mathcal{A}^{n}\cong\bigoplus_{i}\operatorname{im}(P_{i})$. By Kaplansky’s
theorem, we have that each $\operatorname{im}(P_{i})$ is a free submodule. Now
take any basis for each $\operatorname{im}(P_{i})$, put the column vectors
together, and then use the polar decomposition [15] trick to arrive at the
multiplicative decomposition $H=[U_{1}\mid U_{2}\mid\ldots\mid
U_{k}](E_{1}\oplus E_{2}\oplus\ldots\oplus E_{k})[U_{1}\mid
U_{2}\mid\ldots\mid U_{k}]^{\ast}$.

The same can be done for the $Q_{i}$s to arrive at $H=[V_{1}\mid
V_{2}\mid\ldots\mid V_{k}](F_{1}\oplus\ldots\oplus F_{k})[V_{1}\mid
V_{2}\mid\ldots\mid V_{k}]^{\ast}$.

Imagine for the sake of contradiction that $E_{1}$ is not unitarily similar to
any $F_{i}$. This then produces the decomposition
$\mathcal{A}^{n}\cong\bigoplus_{i}(\operatorname{im}(Q_{i})\cap\operatorname{im}(P_{1}))\oplus\bigoplus_{i}(\operatorname{im}(Q_{i})\cap\operatorname{im}(P_{1})^{\perp})$,
which has more than $k$ non-zero factors. This contradicts the maximality of
$k$. So there must be some $F_{i}$ unitarily similar to $E_{1}$. Assume
without loss of generality that this is $F_{1}$. Use the cancellation property
to cancel $E_{1}$ and $F_{1}$ and arrive at the fact that $\bigoplus_{i\geq
2}E_{i}$ is unitarily similar to $\bigoplus_{i\geq 2}F_{i}$. In a similar way,
we conclude that each $E_{j}$ is unitarily similar to some $F_{i}$. The
conclusion follows. $\Box$

###### Proposition 1

The above conjecture is true when the underlying algebra of $\mathcal{A}$ is
the ring of $n\times n$ matrices $\mathcal{M}_{n}(\mathcal{D})$over any
division $\ast$-algebra $\mathcal{D}$.

Proof  Either $\mathcal{D}$ is:

  * $\bullet$

The real numbers with the trivial involution.

  * $\bullet$

The complex numbers with either the involution
$\operatorname{id}_{\mathbb{C}}$ or $(-)^{\ast}$.

  * $\bullet$

The quaternions with either the involution
$t+x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k}\mapsto
t-x\boldsymbol{i}-y\boldsymbol{j}-z\boldsymbol{k}$ or
$t+x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k}\mapsto
t-x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k}$. Note that while it is true
that the quaternions have infinitely many involutions, it admits only two up
to isomorphism of $\ast$-algebras. [16]

We verify the conjecture for each case in turn.

Let $\mathcal{D}$ be the real numbers. The theorem is equivalent to the
spectral theorem here.

Let $\mathcal{D}$ be the complex numbers with the involution $(-)^{\ast}$. The
theorem is equivalent to the spectral theorem here.

Let $\mathcal{D}$ be the complex numbers with the involution
$\operatorname{id}_{\mathbb{C}}$. Note that every square complex matrix is
similar to a complex symmetric matrix. Thus, given a complex-symmetric matrix
$S$, take its Jordan Normal Form, and replace each Jordan block with the
complex symmetric matrix which it’s similar to – giving $S\sim
J_{1}\oplus\ldots\oplus J_{k}$ where each $J_{i}$ is complex-symmetric. We
have that $S$ is similar to another complex-symmetric matrix. But then $S$ is
furthermore orthogonally similar to $J_{1}\oplus\ldots\oplus J_{k}$, by the
polar decomposition trick [15].

Let $\mathcal{D}$ be the quaternions with the standard involution
$t+x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k}\mapsto
t-x\boldsymbol{i}-y\boldsymbol{j}-z\boldsymbol{k}$. The theorem is equivalent
to the spectral theorem here.

Let $\mathcal{D}$ be the quaternions with the non-standard involution
$t+x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k}\mapsto
t-x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k}$. The proof here is the same
as in the case of $\mathbb{C}$ equipped with the involution
$\operatorname{id}_{\mathbb{C}}$. We only need to observe that:

  * $\bullet$

an analogue of the Jordan Normal Form exists, [16]

  * $\bullet$

every square quaternion matrix is similar to a complex-symmetric matrix,

  * $\bullet$

the polar decomposition generalises to this setting. This is presently proved
in a MathOverflow post [20].

$\Box$

###### Lemma 2

Non-singular matrices over the quaternions equipped with the non-standard
involution $t+x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k}\mapsto
t-x\boldsymbol{i}+y\boldsymbol{j}+z\boldsymbol{k}$ admit polar decompositions.

Proof  This would be a corollary of the statement that for every non-singular
matrix $M$ that is Hermitian with respect to the above involution, such a
matrix admits a polynomial $p$ with real coefficients such that $p(M)^{2}=M$.

Let $M$ be a matrix Hermitian with respect to this involution. Consider a
representation of $M$ as an $\mathbb{R}$-matrix $\chi(M)$. We can use the
standard technique for generalising analytic functions to matrices by way of
Hermite interpolation. We would like though for the coefficients of the
interpolating polynomial $p\in\mathbb{C}[z]$ (for which $p(M)^{2}=M$) to be
real numbers, otherwise we might encounter problems with non-commutativity. We
can ensure this whenever $\chi(M)$ has no negative real eigenvalues, by
ensuring that for every congruence

$p(z)\equiv\sqrt{z}\pmod{(z-\lambda)^{n}}$

we have another congruence of the form

$p(z)\equiv\overline{\sqrt{z}}\pmod{\left(z-\overline{\lambda}\right)^{n}}.$

This system of congruences becomes contradictory when $\chi(M)$ has a negative
real eigenvalue.

In the event that $M$ has a negative real eigenvalue, we may perturb $M$ in
such a way as to eliminate its real eigenvalues. We may construct a sequence
of approximations $M_{n}$ to $M$, consider the sequence $\sqrt{M_{n}}$, refine
this to a convergent subsequence if need be (by way of Bolzano-Weierstrass)
and then take the limit to obtain a square root of $M$. $\Box$

A corollary of the Wedderburn-Malcev theorem is that every local finite-
dimensional $\mathbb{R}$-algebra admits a vector space decomposition $A\oplus
B$ where $A\in\\{\mathbb{R},\mathbb{C},\mathbb{H}\\}$, and $B$ consists only
of nilpotent elements.

###### Definition 2

Call an involution $(-)^{\ast}$ of a local finite-dimensional
$\mathbb{R}$-algebra $\mathcal{A}$ standard if over the subalgebra $A$ (which
is a division algebra) of $\mathcal{A}=A\oplus B$, we have that $z^{2}=-1$
implies $z^{\ast}=-z$.

Call any other involution non-standard. Notice that the only standard
involution for $\mathbb{C}$ is $a+bi\mapsto a-bi$, while the only non-standard
involution is $a+bi\mapsto a+bi$. Both involutions for the algebra of dual
numbers are standard.

###### Lemma 3

Every unit element $x$ over a local finite-dimensional $\ast$-algebra
$\mathbb{\mathcal{A}}$ over $\mathbb{R}$ that carries a standard involution
admits a square root and a polar decomposition.

Proof  We only need to verify the existence of a square root. Going from the
existence of a square root to a polar decomposition is a fairly standard
argument.

Consider an $\mathbb{R}$-matrix $M$ representing $x^{\ast}x$. We perform
Hermite interpolation to obtain a polynomial $p$ with only real coefficients
for which $p(M)^{2}=M$. To ensure the coefficients of $p$ are real, the
interpolation problem should be set up so that the interpolation points are
complex conjugates of each other. This is only possible to do if $M$ has no
negative eigenvalues. But $M$ can’t have a negative eigenvalue because
$x^{\ast}x\operatorname{mod}J(\mathcal{A})$ (where $J(\mathcal{A})$ is the
Jacobson radical of $\mathcal{A}$) is a positive definite element of
$M_{n}(\mathcal{A}/J(\mathcal{A}))$.

We see now that $p(x^{\ast}x)^{2}=x$. $\Box$

###### Proposition 2

The above conjecture is true when the underlying algebra of $\mathcal{A}$ is
$n\times n$ matrices over a local ring $\mathcal{B}$, and the involution is
standard.

Proof  By the Krull-Schmidt theorem, there exists a decomposition:

$H=\sum_{i=1}^{k}P_{i}HP_{i},$

with

  * $\bullet$

$P_{i}\in\mathcal{M}(\mathcal{A}),$

  * $\bullet$

$P_{i}P_{j}=\delta_{ij}P_{i},$

  * $\bullet$

$P_{i}\neq 0$.

but not necessarily with $P_{i}^{\ast}=P_{i}$. We seek to make this last
identity true as well.

Choose some $i\in\\{1,2,\ldots,k\\}$. Since $\mathcal{A}$ is a local ring, we
may choose a basis $\\{v_{1},v_{2,}\ldots\\}$ where each $v_{j}$ belongs to
the image of $P_{i}$. We now show how to construct an improved basis
$\\{v^{\prime}_{1},v^{\prime}_{2},\ldots\\}$ of $\operatorname{im}(P_{i})$
that is orthonormal.

We choose $v^{\prime}_{1}$ to equal $v_{1}\sqrt{v_{1}^{\ast}v_{1}}^{-1}$. This
definition should make sense as long as $v_{1}^{\ast}v_{1}$ is a unit, because
we know that every unit has a square root. Assume for the sake of
contradiction that it isn’t a unit. Then quotienting by the Jacobson radical,
and using the property of standard involutions, gives that each component of
$v_{1}$ is a non-unit. But then the module spanned by $v_{1}$ is a projective
module which isn’t free, which contradicts Kaplansky’s theorem. So we conclude
that $v_{1}^{\ast}v_{1}$ is indeed a unit and the definition of
$v^{\prime}_{1}$ makes sense.

We now must choose a value of $v^{\prime}_{2}$ to take the place of $v_{2}$.
To this end, consider the linear map
$f:\operatorname{im}(P_{i})\rightarrow\operatorname{im}(P_{i}),x\mapsto
v^{\prime}_{1}{v^{\prime}_{1}}^{\ast}x$. Observe that $f(f(x))=f(x)$ for all
$x$. From this, observe that for every vector $x$, we have that
$x=f(x)-(x-f(x))$, where $x-f(x)$ is in the kernel of $f$. Assume that we have
an $x\in\operatorname{im}(f)\cap\ker(f)$. Then we have $x=f(y)$ for some $y$,
and then $x=f(y)=f(f(y))=f(x)=0$, so $x=0$. Therefore we have the
decomposition $\operatorname{im}(P_{i})=\ker(f)\oplus\operatorname{im}(f)$.
Since $\mathcal{A}$ is local, $\ker(f)$ is a free module, whose every element
is orthogonal to $v^{\prime}_{1}$. Continuing in the obvious way produces an
orthonormal basis for $\operatorname{im}(P_{i})$.

Putting these orthonormal bases for $\operatorname{im}(P_{i})$ for each $i$
together gives that $H$ is similar to a self-adjoint matrix.

We now use the polar decomposition trick to make $H$ unitarily similar to a
self-adjoint matrix. $\Box$

##  5 Suggestion for programme

We propose a programme, which we think might be useful in applications:

  * $\bullet$

Prove the conjecture above.

  * $\bullet$

Classify the complexity of the corresponding decompositions or canonical
forms. It’s common to use the three-way label domestic, tame and wild. [3, 2]

  * $\bullet$

Investigate the use of polymorphism in programming languages to write the same
numerical algorithm for multiple decompositions. This has some resemblance to
the well-known possibility of using polymorphism in dynamic programming
algorithms [19]. We might limit the numerical algorithms to all those
decompositions of low enough complexity. Are many numerical algorithms simply
the QR algorithm [6] in disguise, written in a polymorphic way?

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