On the spectral structure of Jordan-Kronecker products of symmetric and skew-symmetric matrices

TL;DR

This paper investigates the eigenvalue interlacing properties of Jordan-Kronecker products of symmetric and skew-symmetric matrices, proving new results for low-rank cases and identifying where conjectures hold or fail.

Contribution

It proves eigenvalue interlacing for matrices with rank at most two and for small dimensions, and clarifies the structure of eigenvectors associated with extreme eigenvalues.

Findings

Interlacing holds for symmetric/skew-symmetric matrices with rank ≤ 2.

Interlacing is valid for all pairs when n ≤ 3.

Conjectures fail for larger matrices with rank ≥ 3.

Abstract

Motivated by the conjectures formulated in 2003 by Tun\c{c}el et al., we study interlacing properties of the eigenvalues of $A\otimes B + B\otimes A$ for pairs of $n$-by-$n$ matrices $A, B$. We prove that for every pair of symmetric matrices (and skew-symmetric matrices) with one of them at most rank two, the \emph{odd spectrum} (those eigenvalues determined by skew-symmetric eigenvectors) of $A\otimes B + B\otimes A$ interlaces its \emph{even spectrum} (those eigenvalues determined by symmetric eigenvectors). Using this result, we also show that when $n \leq 3$, the odd spectrum of $A\otimes B + B\otimes A$ interlaces its even spectrum for every pair $A, B$. The interlacing results also specify the structure of the eigenvectors corresponding to the extreme eigenvalues. In addition, we identify where the conjecture(s) and some interlacing properties hold for a number of structured matrices. We settle the conjectures of Tun\c{c}el et al. and show they fail for some pairs of symmetric matrices $A, B$, when $n\geq 4$ and the ranks of $A$ and $B$ are at least $3$.