Graduate Trace of the inverse of matrix products

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SUMMARY

The discussion centers on proving the inequality Trace{[(A*B)^{H}*(A*B)]^{-1}} < Trace{[(A)^{H}*(A)]^{-1}} for matrices A and B with complex-valued zero-mean Gaussian entries that are mutually independent. The participants confirm that this result holds for any pair of matrices A and B, emphasizing the need for a rigorous proof applicable to generally random matrices with independent complex-valued Gaussian distributions. The notation used includes the Hermitian transpose (H) and matrix inverse (-1), which are critical in the context of matrix analysis.

PREREQUISITES
  • Understanding of matrix operations, specifically Hermitian transpose and matrix inverse.
  • Familiarity with Gaussian distributions, particularly complex-valued zero-mean Gaussian entries.
  • Knowledge of linear algebra concepts related to matrix traces and their properties.
  • Experience with probability theory as it applies to random matrices.
NEXT STEPS
  • Research the properties of matrix traces in linear algebra.
  • Study the implications of the inverse of products of matrices in relation to Gaussian random variables.
  • Explore the theory of random matrices, focusing on independent complex-valued Gaussian distributions.
  • Learn about the application of inequalities in matrix theory, particularly in the context of trace inequalities.
USEFUL FOR

Mathematicians, statisticians, and researchers in fields involving linear algebra and random matrix theory, particularly those interested in the properties of Gaussian random matrices and their applications.

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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