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Marginal PDF of the eigen values of a Wishart matrix?

Posted 2 months ago
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Hello,

The elements of a matrix H are Gaussian with 0 mean and unit variance. The Matrix W = H H[ConjugateTranspose] is a Wishart Matrix. I am trying to find the margin PDF of the eigen values of this Wishart Matrix. The size of the matrix is a variable. I would like to start small with 2 * 2 and slowly move up till 32 * 32.

In the Wolfram example

RandomVariate[WishartMatrixDistribution[10, {{1, 1/3}, {1/3, 1}}]]

I am not able to understand what {{1, 1/3}, {1/3, 1}} represents?

Would someone help me as to what values parameters should I enter for the problem I have?

POSTED BY: P V
Answer

That is a matrix that represents scale matrix parameter of the wishart distribution. This is the covariance matrix of one row of H, so in your case, it would be the identity matrix of size equal to the number of columns of H. The first parameter of the Wishart distribution is the degrees of freedom, in your case it’s the number of rows of H.

You won’t be able to directly compute the PDF of the eigenvalues though, as the MatrixPropertyDistribution doesn’t work with PDF. The best you can do is simulate using RandomVariate and create a KernelDensityEstimate.

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