# 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.

## Key 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.