## Inverse, dot product and SVD on matrix-valued random variables

Hi.

Firstly this topic involves both linear algebra and statistics so I'm not sure what this falls under, so sorry if this is not the right forum for this topic.

I have a matrix-valued random variable, C. The elements within each instance of C are dependant, but the elements across samples are independent with known (say Gaussian) distributions with known means and variances:

$C_{ij} \sim N(\mu_{C_{ij}},\sigma^2_{C_{ij}})$

I have another variable, v, which is ( a rather complicted) function of C, such that:

$v=\lambda(W^TCW)$

where

$W=(L^TC^{-1}L)^{-1}L^TC^{-1}$

$\lambda$ is the largest singular value and L is a matrix constant.

I want to find an analytical expression for the mean and variance of v. So to break this down, what i fundamentally need to know is what happens to a matrix-valued random distributions during a dot product, inverse and SVD.

Currently i am generating several instances C and calculating v for each one and working out the moments of v from that, but this takes a long time as the matrices are quite large, so an analytical expression will save me a lot of time.

Is this possible? Any help would be greatly appreciated.

Mark