Spectrally arbitrary patterns
type: Item
instance of: scholarly article
title: Spectrally arbitrary patterns
zbMATH Open document ID: 0957.15012
DOI: 10.1016/S0024-3795(00)00026-4
author: Charles R. Johnson
author: D. D. Olesky
author: John H. Drew
author: P. van den Driessche
published in: Linear Algebra and its Applications
publication date: 2001-03-29T00:00:00Z
review text: An \(n\times n\) sign pattern matrix \(S\) is an inertially arbitrary pattern (IAP) if each nonnegative triple \((n_1,n_2,n_3)\) with \(n_1+ n_2+n_3=n\) is the inertia of a matrix with sign pattern \(S\). Analogously, \(S\) is a spectrally arbitrary pattern (SAP) if, for any given real monic polynomial \(r(x)\) of order \(n\), there is a matrix with sign pattern \(S\) and characteristic polynomial \(r(x)\). Focusing on tree sign patterns, consider the \(n \times n\) tridiagonal sign pattern \(T_n\) with each superdiagonal entry positive, each subdiagonal entry negative, the \((1,1)\) entry negative, the \((n, n)\) entry positive, and every other entry zero. It is conjectured that \(T_n\) is an IAP. By constructing matrices \(A_n\) with pattern \(T_n\), it is proved that \(T_n\) allows any inertia with \(n_3\in \{0,1,2, n-1,n\}\) for all \(n\geq 2\). This leads to a proof of the conjecture for \(n\leq 5\). The truth of the conjecture is extended to \(n\leq 7\) by showing the stronger result that \(T_n\) is a SAP. Further questions about patterns that are SAPs and IAPs are considered.
Mathematics Subject Classification ID: 15B35
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 1446347
zbMATH Keywords: characteristic polynomials
zbMATH Keywords: inertia
zbMATH Keywords: nilpotent matrix
zbMATH Keywords: inequalities involving eigenvalues
zbMATH Keywords: sign pattern matrix
zbMATH Keywords: tree sign patterns
zbMATH Keywords: tridiagonal sign pattern
reviewed by: Rodica Covaci
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1446347