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 ⊗ In × n+In × n ⊗ B+Θ ⊗ Ξ ∈ Cn² × n², where $A,B$ are independent Wigner matrices and Θ,Ξ 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 min_X ∈ Rn × n 1/2\|XA+BX\|F²+1/2∑ᵢⱼ ξᵢθⱼ xᵢⱼ² 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 → ∞.