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Let $\Sigma$ be a $n \times n$ matrix, $V$ a $2 \times 2$ matrix and $I_{2 n}$ the identity matrix on dimension $2n \times 2n$. Both $\Sigma$ and $V$ are covariance matrices, thus real, symmetric and positive definite.

I need to calculate $(\Sigma\otimes V+\phi I_{2 n})^{-1}$ where $\phi$ is a positive scalar and $\otimes$ is the Kronecker product. How can I use the property of the Kronecker product to compute the inversion efficiently?

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  • $\begingroup$ Thx. That could have an impact on possible approaches, so I took the liberty of editing your question to include this info. I'll delete my comment and suggest you do so, too, to clean up. $\endgroup$ – Stephan Kolassa Jun 12 '14 at 9:28
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    $\begingroup$ This might lead nowhere, but is a bit like evaluating 1/(1+x). What happens if you expand in a Taylor series? $\endgroup$ – Paul Jun 12 '14 at 9:32
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    $\begingroup$ At least for the second term, it's a trivial special case of the Sherman–Morrison formula for matrix inversion. $\endgroup$ – user1963 Jun 12 '14 at 9:36
  • $\begingroup$ Do you need the result for pencil-paper work or for implementation in a statistical algorithm? For the latter case, you don't need to calculate the matrix inverse. It is no case known where a matrix inversion is inevitable. $\endgroup$ – Horst Grünbusch Jun 12 '14 at 10:40
  • $\begingroup$ I need it for a statistical algorithm.... $\endgroup$ – niandra82 Jun 12 '14 at 10:46
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You may note that $\ker(\Sigma \otimes V + \phi I_{2n}) = \{0\}$ if and only if $\phi$ is not in the spectrum of $\Sigma \otimes V$. It follows that $(\Sigma \otimes V + \phi I_{2n})$ is invertible if and only if $\phi $ is not an eigenvalue of $\Sigma \otimes V$. The spectrum of a Kronecker product of matrix is already studied and you can express it explicitly in termes of the spectrum of $A$ and the spectrum of $V$. Here it is shown that if $(\lambda,\sigma)$ and $(\mu,v)$ are two eigenpairs of $\Sigma$ and $V$ respectively, then $(\lambda\mu,\sigma \otimes v)$ is an eingenpair of $ \Sigma \otimes V$. Anyway in the link you should find some interesting factorization for solving efficitently the linear equation system $(\Sigma \otimes V + \phi I_{2n})x=b$. Computing the inverse directly is not very efficient except unless you need to solve this system a large amount of times for fixed $V$ and $\Sigma$ and varying $b$.

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