About the matrix function X->AX+XA

TL;DR

This paper investigates properties of the matrix function X -> AX + XA over different fields, establishing conditions for invertibility and sign of determinants based on matrix rank and spectral properties.

Contribution

It provides new results on the invertibility of A+B when B is similar to A and characterizes the sign of det(AX+XA) for even n in relation to A's rank and spectral properties.

Findings

Existence of B similar to A making A+B invertible when rank(A) >= n/2.

Characterization of when det(AX+XA) >= 0 for even n, based on A's rank and A^2.

Conditions linking matrix spectral properties to the sign of the determinant of the matrix function.

Abstract

Let K be an infinite field such that its characteristic is not 2. We show that, for every $A\in\mathcal{M}_n(K)$ such that $\mathrm{rank}(A)\geq n/2$, there exists $B\in\mathcal{M}_n(K)$ such that $B$ is similar to $A$ and $A+B$ is invertible. Let $K$ be a subfield of $\mathbb{R}$. We show that, if $n$ is even, then for every $X\in\mathcal{M}_n(K)$, $\det(AX+XA)\geq 0$ if and only if either $\mathrm{rank}(A)<n/2$ or there exists $\alpha\in K,\alpha\leq 0$, such that $A^2=\alpha I_n$.