# Density function for the trace of a singular value matrix

1. Oct 12, 2013

### Squatchmichae

Hi All,
It's been years since I have re-visited PF.
I have an interesting problem today. I arises in a physical hypothesis testing problem:

Problem Statement: what's the density function for the sum of singular values (trace of the singular value matrix) for a square, Gaussian matrix?

My Approach thus far:

Suppose A is an NxN matrix whose elements are Gaussian R.V.s Under the assumption of mutual independence:

|| A ||$^{2}_{F}$ = || D ||$^{2}_{F}$

where D is the eigenvalue value matrix of A. Because || A ||$^{2}_{F}$ is a sum of the IID Gaussian squares along the diagonal, it it Chi-square:

|| A ||$^{2}_{F}$ ~ $\chi$$^{2}$

where the effective degrees of freedom are N$^{2}$, since the Frobenius norm-squared is tr( A$^{T}$A ), and means we are summing inner products along a diagonal, making N$^{2}$ terms contributing to the sum in total.

I would like to find the seemingly related distribution for tr( S )--that is, the trace of the singular value matrix, S, which has square-rooted eigenvalues.

A naive approach would be to suggest that tr( S ) is a sum of Chi random variables, assuming each singular value is separately also a $\chi$$^{2}$ by Cochran's Theorem.

I am not a mathematician or EE, so if you have insight, I probably will get lost if said insight involves Lie algebras!

Thanks for reading this, and any insight you have is appreciated. This distribution is associated with a detection problem at Los Alamos, and is currently considered quite important for establishing false alarm and detection probabilities of the associated detection statistic.