On the corners of certain determinantal ranges
type: Item
instance of: scholarly article
title: On the corners of certain determinantal ranges
zbMATH Open document ID: 1126.15006
author: Alexander Kovačec
author: Natalia Bebiano
author: João Da Providência
published in: Linear Algebra and its Applications
publication date: 2007-09-13T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/4598
review text: Let \(A\) be a complex \(n\times n\) matrix and \(\text{SO}(n)\) be the group of \(n\times n\) real orthogonal matrices of determinant \(1\). The authors consider the set \(\Delta(A)\subseteq\mathbb{C}\) consisting of all \(\det(A\circ Q)\) (\(Q\in \text{SO}(n)\)) where \(\circ\) denotes the Hadamard product. For each permutation \(\sigma\in S_{n}\) they define \(z_{\sigma}:=\prod_{i=1}^{n} a_{i\sigma(i)}\), and note that \(z_{\sigma}=\det(A\circ Q)\) for suitable signed permutation matrices \(Q\in \text{SO}(n).\) A (weakened) version of the Oliveira Marcus conjecture [see \textit{G. N. De Oliveira}, Linear Multilinear Algebra 12, 125--138 (1982; Zbl 0485.15006)] states that \(\Delta(A)\) is contained in the convex hull of the \(z_{\sigma}\) (\(\sigma\in S_{n}\)). The present authors provide some evidence towards the truth of this conjecture by showing that the local shape of \(\Delta(A)\) at \(z_{\sigma}\) is a truncated cone whose angle at \(z_{\sigma}\) equals the angle \(\widehat{z_{\sigma\tau}z_{\sigma}z}_{\sigma\tau^{\prime}}\) for some transpositions \(\tau,\tau^{\prime}\).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 5190763
zbMATH Keywords: determinantal range
zbMATH Keywords: Hadamard product
zbMATH Keywords: Oliveira-Marcus conjecture
MaRDI profile type: Publication
OpenAlex ID: W2083809509
cites work: Q3704110
cites work: Q3281164
cites work: Bounds for the Determinant of the Sum of Hermitian Matrices
cites work: Q4040931
cites work: A bound for the determinant of the sum of two normal matrices
cites work: Q5522742
cites work: Q5665390
cites work: Q5732721
DOI: 10.1016/J.LAA.2007.04.010
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description: scientific article; zbMATH DE number 5190763