Congruences of a square matrix and its transpose

TL;DR

This paper proves that any square matrix over a field is congruent to its transpose under any nonidentity involution, extending known results to more general involutions.

Contribution

The paper generalizes the congruence relationship between a matrix and its transpose to include all nonidentity involutions on the field.

Findings

Matrices are congruent to their transpose under any nonidentity involution.

The result extends classical congruence properties to broader involutions.

Provides a unified view of matrix congruences over various fields.

Abstract

It is known that any square matrix over any field F is congruent to its transpose. We show that they are also *congruent with respect to any nonidentity involution on F.