A matrix of permanents
type: Item
instance of: scholarly article
title: A matrix of permanents
zbMATH Open document ID: 0533.15006
DOI: 10.1016/0024-3795(84)90113-7
published in: Linear Algebra and its Applications
publication date: 1984-01-01T00:00:00Z
review text: If A is an \(n\times(n-2)\) matrix let \(\bar A\) be the \(n\times n\) matrix with zero diagonal and whose (i,j) element, for \(i\neq j\), is the permanent of the matrix obtained by deleting the \(i^{th}\) and \(j^{th}\) rows of A. If A is positive then \(\bar A\) is nonsingular and has exactly one positive eigenvalue. This fact is used in the proof of the van der Waerden conjecture [cf. \textit{J. H. Van Lint}, ibid. 39, 1-8 (1981; Zbl 0468.15005)]. The author proves that if A is nonnegative then \(\bar A\) is nonsingular if and only if A has no zero submatrix of n-1 lines. Using this he shows that if \((y^ t\bar Ax)^ 2=(y^ t\bar Ay)(x^ t\bar Ax),\) where x and y are column vectors with \(y\geq 0\), then one of the following is true: (i) \(x=\alpha y\) for some real \(\alpha\), (ii) at most one component of y is positive, (iii) A has a zero submatrix of n-1 lines.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 3845758
zbMATH Keywords: matrix of permanents
zbMATH Keywords: Alexandroff inequality
zbMATH Keywords: van der Waerden conjecture
author: R. B. Bapat
reviewed by: Fergus J. Gaines
MaRDI profile type: Publication
full work available at URL: 0024-3795(84)90113-7
OpenAlex ID: W2093013354
cites work: Doubly stochastic matrices with equal subpermanents
cites work: A Permanent Inequality
cites work: Subdefinite Matrices and Quadratic Forms
cites work: Notes on Egoritsjev's proof of the van der Waerden conjecture
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description: scientific article; zbMATH DE number 3845758