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Spectral relations between products and powers of isotropic random matrices
1Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland. zdzislaw.burda@uj.edu.pl
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 2, 2013
Summary
The limiting eigenvalue density of a product of random matrices equals the density of the nth power of a single matrix. This finding simplifies analyzing products of independent non-Hermitian random matrices.
Area of Science:
- Mathematics
- Physics
Background:
- Random matrix theory analyzes properties of matrices with random entries.
- Understanding eigenvalue distributions is crucial in various scientific fields.
Purpose of the Study:
- To establish a relationship between the eigenvalue density of a product of random matrices and the nth power of a single matrix.
- To derive the limiting eigenvalue density for products of independent non-Hermitian random matrices.
Main Methods:
- Analysis of isotropic unitary ensembles in the limit of large matrix size.
- Application of a key observation relating product density to power density.
Main Results:
- Demonstrated that the limiting eigenvalue density of a product of n random matrices equals the density of the nth power of a single matrix.
- Derived the limiting density for products of n independent, identically distributed non-Hermitian matrices with unitary invariant measures.
- Provided evidence for the applicability to isotropic orthogonal ensembles.
Conclusions:
- The established relationship offers a powerful tool for analyzing products of random matrices.
- The findings are applicable to specific examples like products of Girko-Ginibre and truncated unitary matrices.
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