Is This a New Class of Matrices?

TL;DR

This paper introduces a new class of matrices derived from a real square matrix and a sign vector, exploring their algebraic properties, decompositions, and relations to graph components, revealing new structural insights.

Contribution

It defines a novel matrix class via conjugation by signature matrices, analyzes their algebraic structure, and relates their properties to graph connectivity and matrix decompositions.

Findings

Matrices are similar and congruent to the original matrix.

The group of maps forms an abelian group isomorphic to ({Z}_2)^{n-1}.

Number of distinct matrices under the map is 2^{n-t}, linked to graph components.

Abstract

We consider a new class of matrices associated to a real square matrix $A$ and to a vector $\vec{c} \in \{-1,1\}^n$ such that $c_1=1$ by using a map $\varphi_{\vec{c}}$ which turns out to be a conjugation of a matrix $A$ by a signature matrix. It is shown that every such matrix is similar and congruent to a matrix $A$ and that they have same permanental polynomials. There are $2^{n-1}$ maps $\varphi_{\vec{c}}$ and they form an abelian group under the composition of maps isomorphic to the group $({\mathbb{Z}_2}^{n-1}, +)$. A decomposition of matrices, on a symmetric and antisymmetric matrix under a map $\varphi_{\vec{c}}$, is considered. Particularly, it is shown that sum of all principal minors of the order two of a matrix $A$ is equal to the sum of all principal minors of the order two of their symmetric and antisymmetric parts. It is shown that any symmetric matrix and any antisymmetric matrix under the map $\varphi_{\vec{c}}$ are simultaneously permutation similar to certain block matrices which have two blocks. Finally, for a fixed matrix $A$, it is proved that the number of different matrices $\varphi_{\vec{c}}(A)$ is $2^{n-t}$, where $t$ is the number of connected components of the graph $G$ whose adjacency matrix is $A$.