An explicit expression for the minimal polynomial of the Kronecker product of matrices. Explicit formulas for matrix logarithm and matrix exponential

TL;DR

This paper derives explicit formulas for the minimal polynomial of the Kronecker product of matrices using $ ext{P}$-canonical forms, and applies these to matrix functions like exponential and logarithm.

Contribution

It provides a new explicit expression for the minimal polynomial of Kronecker products and derives formulas for matrix exponential and logarithm based on canonical forms.

Findings

Explicit formula for minimal polynomial of Kronecker product

Connection between linear recurrence sequences and minimal polynomials

Formulas for matrix exponential and logarithm derived from canonical forms

Abstract

Using $\mathcal{P}$-canonical forms of matrices, we derive the minimal polynomial of the Kronecker product of a given family of matrices in terms of the minimal polynomials of these matrices. This, allows us to prove that the product $\prod\limits_{i=1}^{m}L(P_{i})$, $L(P_{i})$ is the set of linear recurrence sequences over a field $F$ with characteristic polynomial $P_{i}$, is equal to $L(P)$ where $P$ is the minimal polynomial of the Kronecker product of the companion matrices of $P_{i}$, $1\leq i\leq m$. Also, we show how we deduce from the $\mathcal{P}$-canonical form of an arbitrary complex matrix $A$, the $\mathcal{P}$-canonical form of the matrix function $e^{tA}$ and a logarithm of $A$.