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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 Approachthus far:

SupposeAis anNxNmatrix whose elements are Gaussian R.V.s Under the assumption of mutual independence:

||A||[itex]^{2}_{F}[/itex] = ||D||[itex]^{2}_{F}[/itex]

whereDis the eigenvalue value matrix ofA. Because ||A||[itex]^{2}_{F}[/itex] is a sum of the IID Gaussian squares along the diagonal, it it Chi-square:

||A||[itex]^{2}_{F}[/itex] ~ [itex]\chi[/itex][itex]^{2}[/itex]

where the effective degrees of freedom areN[itex]^{2}[/itex], since the Frobenius norm-squared is tr(A[itex]^{T}[/itex]A), and means we are summing inner products along a diagonal, makingN[itex]^{2}[/itex] 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 isseparatelyalso a [itex]\chi[/itex][itex]^{2}[/itex] 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.

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# Density function for the trace of a singular value matrix

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