compute the inverse of matrix which is the Kronecker product of two vectors

Question

I would like to compute the inverse of the following matrix \begin{equation} A=\begin{pmatrix} a^2b^2+\sigma^2&a^2bd &ab^2c&abcd\\ a^2bd &a^2d^2+\sigma^2&abcd&acd^2\\ ab^2c&abcd&b^2c^2+\sigma^2&bc^2d\\ abcd&acd^2&bc^2d&c^2d^2+\sigma^2 \end{pmatrix} \end{equation} where I can re-write it as $A=\Theta^T\otimes\Theta+\sigma^2I$ and $\Theta$ is given as $$\Theta=\begin{pmatrix} ab\\ ad\\ bc\\ cd \end{pmatrix}.$$ Here $\otimes$ is the outer product. Is there any way to compute the inverse of matrix $A$ based on $\Theta$?

Answer 1

You may use the so-called Sylvester's criterion in order to determine if the symmetric matrix is indeed positive semidefinite or not. See https://en.wikipedia.org/wiki/Sylvester%27s_criterion for how to use such criterion.

Another way would be to do the eigenvalue decomposition of the matrix, which exists due to the fact that the matrix is symmetric, and then see if all the eigenvalues are non-negative.