A bound for the determinant of certain Hadamard products and for the determinant of the sum of two normal matrices
type: Item
instance of: scholarly article
title: A bound for the determinant of certain Hadamard products and for the determinant of the sum of two normal matrices
zbMATH Open document ID: 0799.15002
DOI: 10.1016/0024-3795(94)90356-5
published in: Linear Algebra and its Applications
publication date: 1994-11-17T00:00:00Z
review text: Let \(A\) and \(U\) be complex \(n \times n\) matrices and suppose that either \(\sum^ n_{i = 1} | u_{ij} | \leq 1\) for all \(j\), or \(\sum^ n_{j = 1} | u_{ij} | \leq 1\) for all \(i\). The author proves that \(| \text{det} (A \circ U) | < \max \sum_{\sigma \in S_ n} | a_{i \sigma (i)} |\). In fact, he gives two proofs. Scale \(A\) so that \(\max \sum_{\sigma \in S_ n} | a_{i \sigma (i)} | = 1\). In the first proof he shows that \(| \text{det} (A \circ U) | \leq | \text{det} (Z \circ U) |\) for another matrix \(Z\) with each \(| z_{ij} | = 1\) and then he applies the Hadamard determinant inequality to \(| \text{det} (Z \circ U) |\). In the second proof he uses quite different techniques to show there exists \(B\) with \(| b_{ij} | \leq 1\) and \(| \text{det} (A \circ U) | = | \text{det} (B \circ U) |\), and then he applies the Hadamard inequality. An immediate corollary is: when \(M\) and \(N\) are \(n \times n\) and normal, with eigenvalues \(\{\mu_ i\}\) and \(\{\nu_ i\}\) respectively, then \(| \text{det} (M + N) | \leq \max_{\sigma \in S_ n} | \mu_ i + \nu_{\sigma (i)} |\). This is the best possible upper bound one could expect for \(| \text{det} (M + N) |\) for general normal \(M\) and \(N\), and it generalizes an inequality of \textit{J. F. Queiró} and \textit{A. Kovaček} [Linear Multilinear Algebra 33, No. 3/4, 171-173 (1993; Zbl 0766.15023)]. It also supports the determinantal conjecture of \textit{M. Marcus} [Indiana Univ. Math. J. 22, 1137-1149 (1973; Zbl 0255.15020)] and \textit{G. N. Oliveira} [Research problem: Normal matrices. Linear Multilinear Algebra 12, 153-154 (1982)].
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 566105
zbMATH Keywords: determinant
zbMATH Keywords: Hadamard products
zbMATH Keywords: sum of normal matrices
zbMATH Keywords: Hadamard determinant inequality
zbMATH Keywords: upper bound
author: S. W. Drury
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90356-5
OpenAlex ID: W2048429424
cites work: Q3998992
cites work: Q3954445
cites work: Q5654173
cites work: Transversal theory. An account of some aspects of combinatorial mathematics
cites work: A bound for the determinant of the sum of two normal matrices
cites work: Q3293614
cites work: Cones, graphs and optimal scalings of matrices
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description: scientific article; zbMATH DE number 566105