Equivalence and congruence of matrices under the action of standard parabolic subgroups

TL;DR

This paper establishes precise conditions for when matrices are equivalent or congruent under the action of standard parabolic subgroups of GL(n, F), extending understanding of matrix transformations.

Contribution

It provides necessary and sufficient criteria for P-equivalence and P-congruence of matrices under standard parabolic subgroup actions, including symmetric and alternating matrices.

Findings

Characterization of P-equivalence conditions

Criteria for P-congruence of symmetric matrices

Results applicable over arbitrary fields

Abstract

We find necessary and sufficient conditions for $P$-equivalence of arbitrary matrices and $P$-congruence of symmetric and alternating matrices, where $P$ is standard parabolic subgroup of $GL_n(F)$ and $F$ is an arbitrary field.