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Density function for the trace of a singular value matrix

  1. Oct 12, 2013 #1
    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.
  2. jcsd
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