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The Kronecker sum of square matrices combines their Kronecker products with identity matrices. If $Ax=\lambda x$ and $By=\mu y$, then $x\otimes y$ is an eigenvector with eigenvalue $\lambda +\mu$. Tensor-product spatial grids therefore turn separated coordinate Laplacians into a Kronecker sum. Two Hermitian matrices give a Hermitian Kronecker sum, allowing an energy method without computing every eigenvector.

At an interior grid point, the five-point finite difference method givesLetwhere the zero Dirichlet boundary values are already incorporated. Stack the unknowns with the $y$ index varying fastest. Using the Kronecker product, defineThen the semidiscrete equation isMoreover,Thus the directional matrices commute, as in the five-point Dirichlet Laplacian as a Kronecker sum.