# 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 $$V(x,y)=f(x) g(y)$$.