Determinants, permanents and some applications to statistical shape theory
type: Item
instance of: scholarly article
title: Determinants, permanents and some applications to statistical shape theory
zbMATH Open document ID: 1263.15005
author: Francisco J. Caro-Lopera
author: Graciela González-Farías
author: Narayanaswamy Balakrishnan
published in: Journal of Multivariate Analysis
publication date: 2013-01-16T00:00:00Z
review text: Let \(|\mathbf{A}|\) and \([\mathbf{A}]\) denote the determinant and, respectively, the permanent of a square matrix \(\mathbf{A}=(a_{ij})\). From the introduction (with minor changes): Modern computing technology can use existing definitions and algorithms to compute \(|\mathbf{A}|\) and \([\mathbf{A}]\) quickly and efficiently. However, from a theoretical viewpoint, it is of interest to propose representations for \(|\mathbf{A}|\) and \([\mathbf{A}]\) in terms of~\(\mathbf{A}\), instead of the usual representation in terms of the elements~\(a_{ij}\) or submatrices. In this paper, some solutions are provided to these problems. In Section~2, an expansion for~\(|\mathbf{A}|\) as a function of powers of traces, and a formula for integer powers of~\(|\mathbf{A}|\) are presented. They are simplified for positive definite matrices, and are expanded in terms of zonal polynomials. Section~3 uses these formulas to establish a connection between \(|\mathbf{A}|\) and~\([\mathbf{A}]\). Section~4 applies the formulas for powers of~\(|\mathbf{A}|\) to random matrix theory based on elliptical distributions with kernels that depend on determinants. Some integrals, similar to those given in case of traces by \textit{C.~S.~Herz} [Ann. Math. (2) 61, 474--523 (1955; Zbl 0066.32002)] are also derived. Finally, an application to statistical shape theory is discussed by deriving the central nonisotropic configuration density in terms of zonal polynomials.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 11P82
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 6126025
zbMATH Keywords: determinants
zbMATH Keywords: permanents
zbMATH Keywords: zonal polynomials
zbMATH Keywords: statistical shape theory
zbMATH Keywords: elliptical distributions
reviewed by: Jorma K. Merikoski
MaRDI profile type: Publication
full work available at URL: j.jmva.2012.07.008
OpenAlex ID: W2044643142
DOI: 10.1016/J.JMVA.2012.07.008
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description: scientific article; zbMATH DE number 6126025