Multicritical microscopic spectral correlators of hermitian and complex matrices

Congruence of Hermitian Matrices by Hermitian Matrices

Two Hermitian matrices A, B ∈ Mn(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C ∈ Mn(C) such that B = CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible iner...

Perturbations of Hermitian Matrices and Applications to Spectral Symbols

It is often observed in practice that matrix sequences {An}n generated by discretization methods applied to linear differential equations, possess a Spectral Symbol, that is a measurable function describing the asymptotic distribution of the eigenvalues of An. Sequences composed by Hermitian matrices own real spectral symbols, that can be derived through the axioms of Generalized Locally Toepli...

Complex-Valued Neural Network for Hermitian Matrices

This paper proposes neural network for computing the eigenvectors of Hermitian matrices. For the eigenvalues of Hermitian matrices, we establish an explicit representation for the solution of the neural network based on Hermitian matrices and analyze its convergence property. We also consider to compute the eigenvectors of skew-symmetric matrices and skew-Hermitian matrices, corresponding to th...

Hermitian matrices : Spectral coupling, plane geometry/trigonometry and optimisation

The paper presents the information processing that can be performed by a general hermitian matrix when two of its distinct eigenvalues are coupled, such as λ < λ′. Setting a = λ+λ ′ 2 and e = λ′−λ 2 > 0, the new spectral information that is provided by coupling is expressed in terms of the ratios e |a| (if λλ ′ > 0) or |a| e (if λλ ′ < 0) and of the product |a|e. The information is delivered in...

J-spectral factorization of regular para-Hermitian transfer matrices