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Eigenvalue density of correlated complex random Wishart matrices
Steven H Simon1, Aris L Moustakas
1Lucent Technologies, Bell Labs, Murray Hill, New Jersey 07974, USA.
Summary
We precisely calculated the eigenvalue density for correlated Wishart random matrices. This method applies to complex matrices in various scientific fields, including information theory and multivariate analysis.
Area of Science:
- Random matrix theory
- Multivariate statistics
- Information theory
Background:
- Correlated Wishart matrices (M dagger M) are essential in statistical analysis.
- These matrices appear in diverse scientific applications.
- Understanding their eigenvalue density is crucial for data analysis.
Purpose of the Study:
- To exactly calculate the eigenvalue density of a specific class of correlated Wishart random matrices.
- To provide a method applicable to complex matrices with positive definite correlations.
Main Methods:
- Utilized a character expansion method for exact calculation.
- Analyzed random matrices of the form M dagger M.
- Considered complex matrices M from a normalized distribution P(M) ~ exp(-Tr[AMB M dagger]).
Main Results:
- Derived the exact eigenvalue density for correlated Wishart matrices.
- The method is applicable for positive definite matrices A and B of arbitrary dimensions.
Conclusions:
- The character expansion method offers an exact analytical solution for correlated Wishart matrix eigenvalue densities.
- This work provides a valuable tool for fields utilizing these matrices, such as information theory and multivariate analysis.
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