A new invariant under congruence of nonsingular matrices

TL;DR

This paper introduces a novel invariant for nonsingular matrices under congruence transformations, based on the trace of the transpose times the inverse, providing a new tool for matrix classification.

Contribution

The authors propose a new invariant, $Tr(^t extbf{A} extbf{A}^{-1})$, under congruence of nonsingular matrices, expanding the set of tools for matrix analysis.

Findings

The invariant remains unchanged under congruence transformations.

It offers a new perspective for classifying nonsingular matrices.

Potential applications in matrix theory and related fields.

Abstract

For a nonsingular matrix $A$, we propose the form $Tr(^t\!A A^{-1})$, the trace of the product of its transpose and inverse, as a new invariant under congruence of nonsingular matrices.

Equations

$$\sigma {(}^{t}XAX)$$

$$\sigma {(}^{t}A)$$

$$\sigma (A^{-1})$$

$$\sigma (adj(A))$$

$$\sigma (cA)$$

$$\sigma (A)$$

$$\begin{gathered} A^{\prime }=\frac{1}{|X|}{ \\ }^{t}XAX. \end{gathered}$$

$$A(a,b,c)=\left(\begin{array}{ccc}1& a& b\\ 0& 1& c\\ 0& 0& 1\end{array}\right)$$

Full text

000Keywords: Linear algebra, Invariant of matrices, Congruence, Zeropotent algebras0002010 Mathematics Subject Classification: Primary 15A15; Secondary 15A72000Research of the first author is supported in part by JSPS KAKENHI Grant Number JP18K11172.000Research of the second author was supported in part by JSPS KAKENHI Grant Number JP25400120.

1 Introduction

Let $K$ be a field. We denote the set of all $n\times n$ matrices over $K$ by $M(n,K)$, and the set of all $n\times n$ nonsingular matrices over $K$ by $GL(n,K)$. For $A,B\in M(n,K)$, $A$ is similar to $B$ if there exists $P\in GL(n,K)$ such that $P^{-1}AP=B$, and $A$ is congruent to $B$ if there exists $P\in GL(n,K)$ such that ${}^{t}\!PAP=B$, where ${}^{t}\!P$ is the transpose of $P$. A map $f:M(n,K)\rightarrow K$ is an invariant under similarity of matrices if for any $A\in M(n,K)$, $f(P^{-1}AP)=f(A)$ holds for all $P\in GL(n,K)$. Also, a map $g:M(n,K)\rightarrow K$ is an invariant under congruence of matrices if for any $A\in M(n,K)$, $g(^{t}\!PAP)=g(A)$ holds for all $P\in GL(n,K)$, and $h:GL(n,K)\rightarrow K$ is an invariant under congruence of nonsingular matrices if for any $A\in GL(n,K)$, $h(^{t}\!PAP)=h(A)$ holds for all $P\in GL(n,K)$.

As is well known, there are many invariants under similarity of matrices, such as trace, determinant and other coefficients of the minimal polynomial. In case the characteristic is zero, rank is also an example. Moreover, much research on similarity (or simultaneous conjugation) has been done from the viewpoint of invariant theory for a group action on tuples of matrices, for example, see [2]. On the other hand, few invariants under congruence of matrices or nonsingular matrices are known except for rank. Restricted to symmetric matrices over the reals, another exception would be from Sylvester’s law of inertia, which states that the numbers of positive, negative, and zero eigenvalues are all invariants under congruence of real symmetric matrices. In this note, we propose a new invariant under congruence of general nonsingular matrices. The form is $\sigma(A)=Tr(^{t}\!AA^{-1})$, where $Tr$ denotes trace.

2 Invariance and other properties of $\sigma$

First of all, let us show that $\sigma$ is an invariant under congruence of nonsingular matrices. In fact, for any nonsingular matrix $X$,

[TABLE]

As an immediate corollary, any polynomial in $\sigma(A)$ over $K$ is also an invariant under congruence of nonsingular matrices. Moreover, any rational function in $\sigma(A)$ over $K$ is an invariant under congruence of nonsingular matrices for which its denominator does not vanish.

Next, let us describe some other properties of $\sigma$.

Proposition 1**.**

For any $n\times n$ nonsingular matrix $A$ over $K$, the following equalities hold.

1. (1)

$\sigma(^{t}\!A)=\sigma(A)$. 2. (2)

$\sigma(A^{-1})=\sigma(A)$. 3. (3)

$\sigma(adj(A))=\sigma(A)$, where $adj(A)$ is the adjugate of $A$. 4. (4)

$\sigma(cA)=\sigma(A)$* for any $c\in K\setminus\{0\}$.* 5. (5)

$\sigma(A)=n$* if $A$ is symmetric.*

Proof.

(1)

[TABLE]

(2)

[TABLE]

(3)

[TABLE]

(4) For any $c\in K\setminus\{0\}$,

[TABLE]

(5) If ${}^{t}\!A=A$, then

[TABLE]

3 Background on $\sigma$

Let us briefly describe the background that led to the discovery of $\sigma$. The authors of the present note with Sin-Ei Takahasi and Makoto Tsukada tried to classify three-dimensional zeropotent algebras up to isomorphism, where a zeropotent algebra is a nonassociative algebra in which the square of any element is zero, see [1, 3]. We there expressed an algebra by its structure constants. For a three-dimensional zeropotent algebra $A$, let $\{e_{1},e_{2},e_{3}\}$ be a linear base of $A$. By zeropotency of $A$ it suffices to consider the structure constants for $e_{2}e_{3},e_{3}e_{1},e_{1}e_{2}$, namely, the algebra can be identified with the $3\times 3$ matrix $A$ such that ${}^{t}(e_{2}e_{3}\,\,e_{3}e_{1}\,\,e_{1}e_{2})=A\,^{t}(e_{1}\,\,e_{2}\,\,e_{3})$. We hereafter use the same symbol $A$ both for the matrix and for the algebra.

The following proposition gives a criterion for two zeropotent algebras to be isomorphic.

Proposition 2** ([1]).**

Let $A$ and $A^{\prime}$ be three-dimensional zeropotent algebras over $K$. Then, $A$ and $A^{\prime}$ are isomorphic if and only if there is a nonsingular matrix $X$ satisfying

[TABLE]

In particular, if $K$ is algebraically closed, then $A$ and $A^{\prime}$ are isomorphic if and only if there is a nonsingular matrix $X$ satisfying $A^{\prime}=\,^{t}\!XAX$, that is, $A$ and $A^{\prime}$ are congruent.

Our strategy of classification was to divide zeropotent algebras into curly algebras and straight algebras. A zeropotent algebra is curly if the product of any two elements lies in the space spanned by these elements, otherwise straight. Let $K$ be an algebraically closed field. A straight algebra of rank 3 can be expressed by the canonical form

[TABLE]

with $a,b,c\in K$.

During the course of classifying the straight algebras, we encountered the quantity $D=a^{2}+b^{2}+c^{2}-abc$, which seems strange because it is not homogeneous with respect to $a$, $b$, and $c$. That is, in case where $D\neq 0$ and $a^{2}+b^{2}-abc\neq 0$, we showed that $A(a,b,c)$ is congruent to $A(\sqrt{D},0,0)$. From this, it turned out that $D$ is an invariant under congruence of some $A(a,b,c)$-type matrices. Inspired by this fact, we got interested in elucidating the origin of $D$. From $D$, later, by backward reasoning from the canonical form to a general matrix using computer, the second author derived a more general homogeneous quantity $\dfrac{1}{|A|}\{a_{13}^{2}a_{22}+3a_{12}a_{23}a_{31}-a_{21}a_{23}a_{31}+a_{22}a_{31}^{2}+a_{11}(a_{23}-a_{32})^{2}-a_{12}a_{31}a_{32}-a_{21}a_{31}a_{32}-a_{13}(a_{21}a_{23}+2a_{22}a_{31}-3a_{21}a_{32}+a_{12}(a_{23}+a_{32}))+a_{12}^{2}a_{33}-2a_{12}a_{21}a_{33}+a_{21}^{2}a_{33}\}$, for a $3\times 3$ matrix $A=(a_{ij})$ of rank 3. Let us denote this quantity by $\kappa(A)$. By computer we confirmed that $\kappa$ is certainly an invariant under congruence of general nonsingular matrices, but could not prove it mathematically. Since the original expression of $\kappa(A)$ was awkward and complicated, by an enormous amount of transformations of trial and error, the first author obtained the more beautiful form $3-\dfrac{1}{|A|}\displaystyle\sum_{i,j}(a_{ij}\widetilde{a_{ji}})$, where $\widetilde{a_{ij}}$ denotes the $(i,j)$ cofactor, that is, the product of $(-1)^{i+j}$ and the $(i,j)$ minor. Denoting $\dfrac{1}{|A|}\displaystyle\sum_{i,j}(a_{ij}\widetilde{a_{ji}})$ by $\sigma(A)$, by further reasoning he had $\sigma(A)=\dfrac{1}{|A|}Tr(A\,adj(^{t}\!A))$, and finally by using ${}^{t}\!A(adj(^{t}\!A))=|^{t}\!A|E$, reached the simple form $Tr(^{t}\!AA^{-1})$ as initially defined and then $\kappa(A)=3-Tr(^{t}\!AA^{-1})$. Through this expression we can now mathematically prove that $\kappa$ is an invariant under congruence of nonsingular matrices.

Beyond the topic of three-dimensional zeropotent algebras, we found that in general, for any integer $n\geq 1$, $\sigma$ is an invariant under congruence of $n\times n$ nonsingular matrices over a general field.