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

Motivated by the conjectures formulated in 2003 [28], we study interlacing properties of eigenvalues A⊗B+B⊗A for pairs n-by-n matrices A,B. We prove that every pair symmetric (and skew-symmetric matrices) with one them at most rank two, odd spectrum (those determined eigenvectors) interlaces its even eigenvectors). Using this result, also show when n≤3, The results specify structure eigenvectors corresponding to extreme eigenvalues. In addition, identify where conjecture(s) and some hold a number structured matrices. settle [28] they fail A,B, n≥4 ranks A B are least 3.