# Inverse of a diagonal matrix plus a Kronecker product?

Given two matrices $X$ and $Y$, it's easy to take the inverse of their Kronecker product:

$(X\otimes Y)^{-1} = X^{-1}\otimes Y^{-1}$

Now, suppose we have some diagonal matrix $\Lambda$ (or more generally an easily inverted matrix, or one for which we already know the inverse). Is there a closed-form expression or efficient algorithm for computing $(\Lambda + (X\otimes Y))^{-1}$?

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+1 What's the motivation for your question? –  draks ... Jan 30 '13 at 21:37
I have a convex optimization problem I'm trying to solve, and the Hessian of the objective takes the form above. If I can compute the inverse Hessian efficiently I can use Newton's method, which is far preferable to gradient methods. –  David Pfau Jan 30 '13 at 22:36

The matrices corresponding to $X$ and $Y$ in the cited paper are positive semidefinite and $\Gamma$ is a multiple of $I$, but I think what the OP asks for is more general. –  user1551 Feb 27 '13 at 8:49