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For questions about inverses and pseudoinverses of matrices.
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How to calculate the inverse of the sum of an identity and a Kronecker product efficiently?
Excellent answer by Betrand. But it's easy to verify that the final form should be $K^{-1} = (P\otimes Q)(I + \Lambda_1 \otimes \Lambda_2)^{-1} (P \otimes Q)^T$.
Here is some R code:
A = matrix(c(3, …