On matrices in finite free position

TL;DR

This paper investigates pairs of matrices in finite free position, exploring their characteristic polynomials and algebraic structures, revealing specific classes of matrices that form such pairs under additive and multiplicative convolutions.

Contribution

It characterizes the algebraic sets of matrix pairs in finite free position, identifying key classes of matrices that form these pairs under additive and multiplicative convolutions.

Findings

Identifies diagonal vs. principally balanced matrices as finite free pairs.

Shows upper/lower triangular matrices form finite free pairs with constant diagonal matrices.

Includes scalar matrices vs. all matrices as finite free pairs.

Abstract

We study pairs $(A,B)$ of square matrices that are in additive (resp. multiplicative) finite free position, that is, the characteristic polynomial $\chi_{A+B}(x)$ (resp. $\chi_{AB}(x)$) equals the additive finite free convolution $\chi_{A}(x) \boxplus \chi_{B}(x)$ (resp. the multiplicative finite free convolution $\chi_{A}(x) \boxtimes \chi_{B}(x)$), which equals the expected characteristic polynomial $\mathbb{E}_U [ \chi_{A+U^* BU}(x) ]$ (resp. $\mathbb{E}_U [ \chi_{AU^* BU}(x) ]$) over the set of unitary matrices $U$. We examine the lattice of (non-irreducible) affine algebraic sets of matrices consisting of finite free complementary pairs with respect to the additive (resp. multiplicative) convolution. We show that these pairs include the diagonal matrices vs. the principally balanced matrices, the upper (lower) triangular matrices vs. the upper (lower) triangular matrices with constant diagonal, and the scalar matrices vs. the set of all square matrices.