Simple and exact extreme eigenvalue distributions of finite Wishart matrices

The authors provide compact and exact expressions for the extreme eigenvalues of finite Wishart matrices with arbitrary dimensions. Using a combination of earlier results, which they refer to as the James–Edelman–Dighe framework, not only an original expression for the cumulative distribution functi...

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Bibliographic Details
Published in:IET communications Vol. 9; no. 7; pp. 990 - 998
Main Authors: Zhang, Wensheng, Zheltov, Pavel, Abreu, Giuseppe
Format: Journal Article
Language:English
Published: The Institution of Engineering and Technology 07.05.2015
Subjects:
catalectic properties
CDF
coefficient matrix
Coefficients
Construction
cumulative density function
Determinants
Eigenvalues
eigenvalues and eigenfunctions
exact extreme eigenvalue distribution
Exact solutions
exponential distribution
exponential vector
finite Wishart matrix
Hankel matrices
incomplete gamma function Hankel matrix
James‐Edelman‐Dighe framework
Mathematical analysis
monomial vector
probability density function
simple extreme eigenvalue distribution
Theorems
Vectors (mathematics)
incomplete gamma function Hankel matrix
CDF
cumulative density function
coefficient matrix
Hankel matrices
monomial vector
eigenvalues and eigenfunctions
exact extreme eigenvalue distribution
simple extreme eigenvalue distribution
exponential distribution
James-Edelman-Dighe framework
finite Wishart matrix
exponential vector
catalectic properties
probability density function
ISSN:1751-8628, 1751-8636
Online Access:Get full text
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Summary:The authors provide compact and exact expressions for the extreme eigenvalues of finite Wishart matrices with arbitrary dimensions. Using a combination of earlier results, which they refer to as the James–Edelman–Dighe framework, not only an original expression for the cumulative distribution function (CDF) of the ‘smallest’ eigenvalue is obtained, but also the CDF of the ‘largest’ eigenvalue and the probability density functions of both are expressed in a similar and convenient matrix form. These compact expressions involve only inner products of exponential vectors, vectors of monomials and certain coefficient matrices which therefore assume a key role of carrying all the required information to build the expressions. The computation of these all-important coefficient matrices involves the evaluation of a determinant of a Hankel matrix of incomplete gamma functions. They offer a theorem which proves that the latter matrix has ‘catalectic’ properties, such that the degree of its determinant is surprisingly small. The theorem also implies a closed-form and numerical procedure (no symbolic calculations required) to build the coefficient matrices.
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ISSN:1751-8628
1751-8636
DOI:10.1049/iet-com.2014.0797