Title: Large Non‐Hermitian Random Matrices and Quatartenionic Free Probability Theory
Abstract: This chapter explains (large) non-Hermitian random matrices using the newly developed quatartenionic free probability theory. The most important fact is that non-Hermitian random matrices have complex-valued eigenvalues. The power of the non-Hermitian randommatrix is reminiscent of the power mapping method that was used for noise reduction in the empirical covariance matrix. The chapter combines the contemporary interest in the eigenvalues of large random matrices with the topic of products of random matrices by studying eigenvalue distributions of products of random matrices where the size of the matrices is large. Lyapunov exponents are useful tools that measure the sensitivity of a dynamical system with respect to initial conditions. The chapter also considers two ensembles of random matrices with independent entries. The chapter states the circular law, and defines a class of Hermitian random matrices with independent entries originally introduced by Wigner.
Publication Year: 2017
Publication Date: 2017-02-11
Language: en
Type: other
Indexed In: ['crossref']
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