Invariance of simultaneous similarity and equivalence of matrices under extension of the ground field

TL;DR

This paper provides a new, elementary proof demonstrating that simultaneous similarity and equivalence of matrix families remain invariant when extending the ground field, a significant result especially over finite fields.

Contribution

The paper introduces a simplified proof confirming invariance of simultaneous similarity and equivalence under ground field extension, clarifying a previously complex result.

Findings

Invariance of simultaneous similarity under field extension.

Invariance of simultaneous equivalence under field extension.

Applicable to finite fields and general cases.

Abstract

We give a new and elementary proof that simultaneous similarity and simultaneous equivalence of families of matrices are invariant under extension of the ground field, a result which is non-trivial for finite fields and first appeared in a paper of Klinger and Levy.