Kronecker-product random matrices and a matrix least squares problem

We study the eigenvalue distribution and resolvent of a Kronecker-product random matrix model A I₍ ₍+I₍ ₍ B+ C^n² n², where A, B are independent Wigner matrices and, are deterministic and diagonal. For fixed spectral arguments, we establish a quantitative approximation for the Stieltjes transform by that of an approximating free operator, and a diagonal deterministic equivalent approximation for the resolvent. We further obtain sharp estimates in operator norm for the n n resolvent blocks, and show that off-diagonal resolvent entries fall on two differing scales of n^-1/2 and n^-1 depending on their locations in the Kronecker structure. Our study is motivated by consideration of a matrix-valued least-squares optimization problem ₗ ₑ^₍ ₍ 12\|XA+BX\|F²+12₈₉ ᵢⱼ x₈₉² subject to a linear constraint. For random instances of this problem defined by Wigner inputs A, B, our analyses imply an asymptotic characterization of the minimizer X and its associated minimum objective value as n.