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

Abstract

Motivated by the conjectures formulated in 2003 [28], we study interlacing properties of the eigenvalues of A⊗B+B⊗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 odd spectrum (those eigenvalues determined by skew-symmetric eigenvectors) of A⊗B+B⊗A interlaces its even spectrum (those eigenvalues determined by symmetric eigenvectors). Using this result, we also show that when n≤3, the odd spectrum of A⊗B+B⊗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 [28] and show they fail for some pairs of symmetric matrices A,B, when n≥4 and the ranks of A and B are at least 3.