Partial matrices whose completions have ranks bounded below
type: Item
instance of: scholarly article
title: Partial matrices whose completions have ranks bounded below
zbMATH Open document ID: 1225.15025
author: James McTigue
author: Rachel Quinlan
published in: Linear Algebra and its Applications
publication date: 2011-07-15T00:00:00Z
review text: Let \(A\) be an \(m\times n\) partial matrix over a field~\(\mathbb{F}\). That is, its some entries are specified while the other entries are indeterminates, and specifying them produces a completion of~\(A\). Assuming that \(m=n\) and that \(\mathbb{F}\) has at least \(n+1\) elements, \textit{R.~A.~Brualdi}, \textit{Z.~Huang} and \textit{X.~Zhan} [Linear Algebra Appl.~433, 1452--1462 (2010; Zbl 1205.15042)] proved that if all the completions of~\(A\) have rank~\(n\), then the number of indeterminates in~\(A\) is at most \(\frac{1}{2}n(n-1)\). They also described the case when this bound is attained. The present authors pursue this topic further. Let \(k\leq m,n\) and assume that every completion of~\(A\) has rank at least~\(k\). The authors prove that the number of indeterminates in~\(A\) is at most \(mn-\frac{1}{2}k(k+1)\). They also describe the case when this bound is attained. Their main theoretical tool is a duality relationship between certain affine spaces of matrices.
Mathematics Subject Classification ID: 15A83
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 5924587
zbMATH Keywords: partial matrix
zbMATH Keywords: matrix completion
zbMATH Keywords: rank
zbMATH Keywords: affine space
reviewed by: Jorma K. Merikoski
MaRDI profile type: Publication
full work available at URL: j.laa.2011.04.012
OpenAlex ID: W2011080030
cites work: Singular, nonsingular, and bounded rank completions of ACI-matrices
cites work: Q3831127
cites work: On the matrices of given rank in a large subspace
cites work: On Nilalgebras and Linear Varieties of Nilpotent Matrices, I
cites work: A determinantal version of the frobenius-könig theorem
cites work: ACI-matrices all of whose completions have the same rank
cites work: On two extremal matrix problems
cites work: Spaces of matrices without non-zero eigenvalues in their field of definition, and a question of Szechtman
DOI: 10.1016/J.LAA.2011.04.012
description: scientific article; zbMATH DE number 5924587