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?
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 $b$.