Eigenvalue Statistics of Random Real Matrices

This paper establishes a relationship (Theorem 4.1) between the approaches of A. C. Thompson [8, 9] and E. G. Effros [2] to the representation of simplex algebras, that is, real unital Banach algebras that are simplex spaces with the unit for order identity. It proves that the (nonempty) interior of the associated cone is contained in the principal component of the set of all regular elements of the algebra. It also conjectures that each maximal ideal (in the order sense—see below) of a simplex algebra contains a maximal left ideal of the algebra. This conjecture and other aspects of the relationship are illustrated by considering algebras of n × n real matrices.

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On a class of Jacobi-like procedures for diagonalising arbitrary real matrices

Numerische Mathematik ◽

10.1007/bf01399551 ◽

1979 ◽

Vol 33 (2) ◽

pp. 157-172 ◽

Cited By ~ 11

Author(s):

K. Veselić

Keyword(s):

Real Matrices

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Exact relation between singular value and eigenvalue statistics

Random Matrices Theory and Application ◽

10.1142/s2010326316500155 ◽

2016 ◽

Vol 05 (04) ◽

pp. 1650015 ◽

Cited By ~ 13

Author(s):

Mario Kieburg ◽

Holger Kösters

Keyword(s):

Singular Values ◽

Singular Value ◽

Specific Class ◽

Rotation Invariant ◽

Exact Relation ◽

Finite Matrix ◽

Eigenvalue Statistics ◽

Probability Densities ◽

Analytical Relations ◽

Matrix Spaces

We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint densities of the singular values and the eigenvalues for complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that each of these joint densities determines the other one. Moreover, we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore, we show how to generalize the relation between the singular value and eigenvalue statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.

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