Vanishing minor conditions for inverse zero patterns
type: Item
instance of: scholarly article
title: Vanishing minor conditions for inverse zero patterns
zbMATH Open document ID: 0767.15003
DOI: 10.1016/0024-3795(93)90335-L
author: Charles R. Johnson
author: John S. Maybee
published in: Linear Algebra and its Applications
publication date: 1993-05-16T00:00:00Z
review text: The authors are interested in the following kind of question: Suppose that certain entries of a matrix \(A\) are specified to be zero; what can we deduce from this information alone about vanishing of determinantal minors of \(A\) and \(A^{-1}\)? They term such minors as ``generically'' zero. Their starting point is a classical theorem due to Frobenius (1912): if \(A\) is a \(k\times k\) matrix, then \(A\) has an \(r\times s\) submatrix of zero entries with \(r+s>k\) if and only if \(\text{det }A\) is generically 0. The paper introduces a lot of notation, and the results obtained are rather technical, but we paraphrase one of them (Theorem 4) to give the flavour of the paper. Let \(A\) be an invertible \(n\times n\) matrix, and partition \(\{1,\dots,n\}\) into three nonempty subsets \(\alpha\), \(\beta\) and \(\gamma\). Suppose that the submatrix \(A[\beta\mid\gamma]\) whose rows and columns are indexed by \(\beta\) and \(\gamma\), respectively, has all entries equal to 0. Then for all \(i\in \beta\) and \(j\in\gamma\), we have \(\text{det }A^{-1} [\alpha\cup \{i\}\mid\alpha\cup \{j\}]=0\).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 05C70
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 166231
zbMATH Keywords: inverse zero patterns
zbMATH Keywords: zero-nonzero pattern
zbMATH Keywords: invertible matrix
zbMATH Keywords: vanishing minors
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 166231