I am trying to find an expression for the mean, column-covariance and row-covariance matrices of the product of two matrix-valued Normal distributions.
Here is what I've tried in a special case I need to solve:
Let $G_1$ and $G_2$ be two matrix-valued normal distributions with:
- Mean $\mu_1$ and identity column/row precisions for $G_1$
- $0$ mean and column/row precisions $U^TU$ and $V^TV$ for $G_2$
$G_1$ is equivalent to a multivariate normal $G_1'$ with mean $\mu_1' = vec(\mu_1)$ and identity covariance.
$G_2$ is equivalent to a zero-mean MVN $G_2'$ with precision $V^TV \otimes U^TU$.
It follows that $G_1 G_2$ has covariance $\Sigma = (I + V^TV \otimes U^TU)^{-1}$ and mean $\mu = \Sigma \mu_1'$.
I can then fold-back $\mu$ into its matrix form but I have yet to find an expression for the row and column covariance of the product.
I now try to find two matrices $C$ and $R$ such that $R \otimes C = (I + V^TV \otimes U^TU)$ (I don't think this decomposition always exists).
$U^TU$ and $V^TV$ are symmetric real-valued matrices so it follows from the spectral theorem that there exist two orthogonal bases $P_U$ $P_V$ such that:
$U^TU = P_U \Lambda_U P_U^T$ and $V^TV = P_V \Lambda_V P_V^T$, using the compatibility of the Kronecker with the eigenvalue decomposition and its inversion/transposition properties we get that:
$(I + V^TV \otimes U^TU) = (P_V \otimes P_U)(I + \Lambda_V \otimes \Lambda_U)(P_V \otimes P_U)^T$
and the inverse of $(I + V^TV \otimes U^TU)$ is
$(P_V \otimes P_U)(I + \Lambda_V \otimes \Lambda_U)^{-1}(P_V \otimes P_U)^T$
but I'm stuck at this point.
Any help is welcome.