Mixed discriminants of positive semidefinite matrices
type: Item
instance of: scholarly article
title: Mixed discriminants of positive semidefinite matrices
zbMATH Open document ID: 0696.15007
DOI: 10.1016/0024-3795(89)90009-8
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: If \(A^ k=(a^ k_{ij})\) are \(n\times n\) complex matrices \(k=1,2,...,n\), then their mixed discriminant \(D(A^ 1,...,A^ n)\) is \(\frac{1}{n!}\sum_{\sigma \in S_ n}\det (a_{ij}^{\sigma (j)})\), where \(S_ n\) is the symmetric group of degree n. If all the \(A^ k\) are equal this turns out to be det A, whereas if each \(A^ k\) is a diagonal matrix the mixed discriminant equals \(\frac{1}{n!}per(a^ j_{ii})\). The author gives several alternative ways of defining \(D(A^ 1,...,A^ n)\) and he gives a simple proof of a result of \textit{J. S. Lomont} and \textit{M. S. Cheema} [Linear, Multilinear Algebra, 14, 199-223 (1979; Zbl 0521.15018)] which gives for mixed discriminants an analogue of Ryser's formula for the permanent. He expresses the mixed discriminant as an inner product and derives a Cauchy-Binet formula. He also proves, for nonnegative definite Hermitian matrices, a generalization of König's theorem on 0-1 matrices. He proves that if \(A^ 1,...,A^ n\) are \(n\times n\) nonnegative definite Hermitian matrices, each having trace 1, with \(A^ 1+...+A^ n=I\) then \(D(A^ 1,...,A^ n)>0\) and he gives a partial characterization of the extreme points of the set of n-tuples \((A^ 1,...,A^ n)\) with these properties, in an attempt to get an analogue of the Birkhoff-von Neumann theorem for doubly stochastic matrices.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 15B51
zbMATH DE Number: 4139938
zbMATH Keywords: positive semidefinite matrices
zbMATH Keywords: determinant
zbMATH Keywords: mixed discriminant
zbMATH Keywords: permanent
zbMATH Keywords: Cauchy-Binet formula
zbMATH Keywords: 0-1 matrices
zbMATH Keywords: extreme points
zbMATH Keywords: Birkhoff- von Neumann theorem
zbMATH Keywords: doubly stochastic matrices
author: R. B. Bapat
reviewed by: Fergus J. Gaines
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90009-8
OpenAlex ID: W2016724581
cites work: Inequalities for mixed Schur functions
cites work: A multilinearity property of determinant functions
cites work: Q3880849
cites work: Inequalities for the permanent function
cites work: Permanents
cites work: Transversal theory. An account of some aspects of combinatorial mathematics
cites work: Schur complements and statistics
cites work: Q3729091
cites work: ON SOME PROPERTIES OF MIXED DISCRIMINANTS
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description: scientific article; zbMATH DE number 4139938