linear algebra – Eigenvalues of the sum of a Kronecker product (of two diagonal matrices) with a Kronecker sum [A⊕B+D1⨂D2=A⨂I+I⨂B=D1⨂D2]


We want to find the eigenvalues of the $(n^2 times n^2)$ matrix M, which is the sum of a Kronecker sum and a Kronecker product, that is
$$
M = A oplus B + D_1 otimes D_2 = A otimes I_n + I_n otimes B + D_1 otimes D_2,
$$

where $A,B,D_1,D_2 in mathbb{R}^{n times n}$. $A,B$ are both symmetric and positive definite while $D_1,D_2$ are both diagonal with $D_1=text{diag}{{d_{1,1},dots,d_{1,n}}},D_2=text{diag}{{d_{2,1},dots,d_{2,n}}}$. If we know all spectral properties (eigenvalues, eigenvectors) of $A,B,D_1,D_2$ ($D_1,D_2$ trivial), can we derive the eigenvalues of the matrix $M$?


In practise, this problem results from the eigenvalue problem: Find $(u,lambda)$ such that
$$
mathcal{L} u(x,y) := – Delta u(x,y) + f(x)g(y) u(x,y) = lambda u(x,y) text{ on } Omega=(0,L_x) times (0,L_y)
$$

using a structured Finite Difference Discretization with zero Dirichlet boundary conditions. The linear operator $mathcal{L}$ is a Schrödinger operator with the potential $V(x,y)=f(x) g(y)$.