Graduate Trace of the inverse of matrix products

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The discussion revolves around proving that the trace of the inverse of the product of two independent complex-valued zero-mean Gaussian matrices, A and B, is always less than the trace of the inverse of A's Hermitian product. The original poster seeks clarification on whether this result holds for any random matrices A and B with the specified properties. Participants emphasize the need for a rigorous mathematical approach to demonstrate this inequality. The conversation highlights the complexities involved in dealing with random matrices and their statistical properties. Overall, the inquiry seeks a deeper understanding of matrix behavior in the context of Gaussian distributions.
nikozm
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Hello,

I am puzzled about the following condition. Assume a matrix A with complex-valued zero-mean Gaussian entries and a matrix B with complex-valued zero-mean Gaussian entries too (which are mutually independent of the entries of matrix A).

Then, how can we prove that Trace{[(A*B)^{H}*(A*B)]^{-1}} is always lower that Trace{[(A)^{H}*(A)]^{-1}} ?

The superscripts {H} and {-1} denote the Hermitian transpose and matrix inverse operator, respectively.

Any idea could be helpful.
Thank you very much in advance.
 
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What is the definition of *always* here? I could pick any matrices A and B, and they would have some probability of being sampled by the gaussians you describe. So is this result intended to just be true for any pair of matrices A and B?
 
Yes. I would like to know if (and how) is this result true for generally random matrices A and B (where their elements are particularly independent complex-valued Gaussian distributed).

Any suggestion could be useful. Thanks in advance.
 

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