# eigenvalues of a product with a diagonal random matrix

I am interested in the eigenvalue spectrum of the following matrix A A=(id+diag random) M Where M is a given matrix with a known eigenvalue spectrum, lets say for simplicity that it is Hermitian. diag random is a diagonal random matrix where each entry is an independent random gaussian number. And id is the identity matrix. For some cases of M, such as M=id this problem is very easy. But if M is more complicated than that there seem to be no clear way to explicitly write down M's spectrum. Any ideas how I could proceeed?

Thanks

W

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Try search terms: "random matrix theory" and "stieltjes transforms" – user4423 May 21 '12 at 23:06
Thanks for the comment about the stieltjes transform. I tried that. Unfortunately, the resulting diagrammatic expansion does not lead to a closed form because all the paths contribute equally, the crossing as well as non-crossing paths. That makes the problem quit hard. I tried writting down the first contributing terms, but other than a perturbative expansion did not get anywhere. – Wunderlich May 21 '12 at 23:16
You should also look up "free convolution" (multiplicative). That'll provide you with the tools to tackle this problem. In short, the densities convolve non-commutatively... – user4423 May 21 '12 at 23:22
The multiplicative free convolution does seem like a promising approach. Could you maybe specify a helpful reference that uses this technique to compute spectra? Maybe a good teaching book or paper? Thanks so much! – Wunderlich May 22 '12 at 0:40
This was "untagged". I've tagged it linear algebra, but I'm not sure that's appropriate. Further tag edits welcome. – Gerry Myerson May 22 '12 at 12:32