Density function for the trace of a singular value matrix

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

where D is the eigenvalue value matrix of A. 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 are N[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, making N[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 is separately also 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.

 

[Jhero]

Cheers!</code>The approach I would take is to use the fact that the trace of a matrix is invariant under a change of basis. In other words, if you have an NxN matrix A, and the singular value decomposition of A is A = USV^T (where U and V are orthogonal matrices and S is the diagonal matrix of singular values), then tr(A) = tr(USV^T) = tr(SV^TU) = tr(S). This means that the distribution of tr(A) is the same as the distribution of tr(S). The distribution of tr(A) can be computed using the properties of the Wishart distribution. The Wishart distribution is the distribution of the sum of squares of independent normally distributed random variables. Since the elements of A are independent normal random variables, the distribution of tr(A) is a Wishart distribution. The distribution of tr(S) can then be computed using the fact that each element of S is the square root of a Wishart random variable. This means that the distribution of tr(S) is the convolution of the distributions of the individual elements of S. The convolution of Wishart distributions can be computed using the properties of the Gamma function. The resulting distribution is a Gamma-distributed random variable. So, in summary, the density function for the sum of singular values (trace of the singular value matrix) for a square, Gaussian matrix is a Gamma-distributed random variable.