The sum and the product of two quadratic matrices

TL;DR

This paper characterizes matrices that can be expressed as sums or products of two matrices, each annihilated by a quadratic polynomial, extending previous results to cases involving irreducible polynomials over any field.

Contribution

It provides a complete characterization of matrices as sums or products of matrices satisfying quadratic polynomial equations, including cases with irreducible polynomials over arbitrary fields.

Findings

Characterization of matrices as sums of matrices annihilated by quadratic polynomials.

Characterization of matrices as products of matrices annihilated by quadratic polynomials.

Extension of known results to cases with irreducible polynomials over arbitrary fields.

Abstract

Let $p$ and $q$ be polynomials with degree $2$ over an arbitrary field $\mathbb{F}$. In the first part of this article, we characterize the matrices that can be decomposed as $A+B$ for some pair $(A,B)$ of square matrices such that $p(A)=0$ and $q(B)=0$. The case when both polynomials $p$ and $q$ are split was already known. In the first half of this article, we complete the study by tackling the case when at least one of the polynomials $p$ and $q$ is irreducible over $\mathbb{F}$. In the second half of the article, we use a similar method to characterize, under the assumption that $p(0)q(0) \neq 0$, the matrices that can be decomposed as $AB$ for some pair $(A,B)$ of square matrices such that $p(A)=0$ and $q(B)=0$.