Analyzing a system, I have faced a problem which is related to Random Matrices and in particular Wishart matrix. The problem is as follows:

Lets assume $\boldsymbol{H}$ is an $m\times n$ random matrix with rows drawn independently from an n-variate normal distribution with zero mean, i.e. $\boldsymbol{H}_{(i)}\sim \mathcal{N}(0,\boldsymbol{\Sigma}_i) = \mathcal{N}(0,\alpha\boldsymbol{I})$. I know $\boldsymbol{W} =\boldsymbol{H}\boldsymbol{H}^H$ is a Wishart matrix distributed as $\mathcal{W}_m(n,\boldsymbol{\Sigma})$ with $n>m$. Now, lets assume $\boldsymbol{\Sigma}$ is different for every row, i.e. $\boldsymbol{H}_{(i)}\sim \mathcal{N}(0,\boldsymbol{\Sigma}_i) = \mathcal{N}(0,\alpha_i\boldsymbol{I})$. Is $\boldsymbol{W} =\boldsymbol{H}\boldsymbol{H}^H$ a wishart matrix? if so, I am interested in the distribution of the eigenvalues of $\boldsymbol{W}$. What is the distribution of $\mathcal{l}^{\text{th}}$ eigenvalue of $\boldsymbol{W}$

I will greatly appreciate any hint or any reference which particularly studies the mentioned problem.

  • $\begingroup$ The answer you accepted is wrong. $W$ is not a noncentral Wishart matrix. It is a sum of independent Wishart matrices. See my answer here for some information. $\endgroup$ – Stéphane Laurent Nov 29 '17 at 10:57

The case when each row has different $\Sigma$ leads to non-central Wishart distribution. About the distribution of eigenvalues of this matrix you can read this tutorial or google up "eigenvalues of non central wishart distribution". There has been a lot of research about random matrices and Wishart distribution in past few years, so you should find a lot of interesting and useful articles there.

Hope it's helpful.

  • $\begingroup$ Wrong. The noncentral Wishart distribution is the case of noncentered $H_{(i)}$ with the same covariance matrix. $\endgroup$ – Stéphane Laurent Nov 29 '17 at 10:59

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