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

Abstract

Kronecker products as well as interlacing properties are very commonly used in matrix theory, operator theory and in their applications. We address conjectures formulated in 2003 [19], involving certain interlacing properties of eigenvalues of (A ⊗ B + B ⊗ A) for pairs of symmetric matrices A and B. We disprove these conjectures in general, but we also identify some special cases where the conjectures hold. In particular, 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) interlace its even spectrum (those eigenvalues determined by symmetric eigenvectors).