As described in this Wikipedia article, a discrete laplacian matrix can be made for a 3D regular grid using Kronecker products. I'd like to use the same methodology for $(n-1)\times n$ matrices of the form

$$ \begin{bmatrix} -1 & 1 \\ & \ddots & \ddots \\ & & -1 & 1 \\ \end{bmatrix} $$

Given 3 of these matrices with sizes appropriate for each dimension, I'd like use Kronecker products to build a matrix for 3D problems. For example, given 3D dimensions of $m\times n \times k$, I'd expect a finite-difference matrix of dimension $(m-1)(n-1)(k-1) \times mnk$.

How do I build this desired matrix using Kronecker products?


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