An inequality for positive definite matrices with applications to combinatorial matrices
type: Item
instance of: scholarly article
title: An inequality for positive definite matrices with applications to combinatorial matrices
zbMATH Open document ID: 0884.15013
author: Michael G. Neubauer
published in: Linear Algebra and its Applications
publication date: 1998-03-10T00:00:00Z
review text: Let \(A\in M_n(\mathbb C)\) be a positive definite matrix. Let \(d,f,f'\) stand for the averages of the diagonal, off-diagonal, and modified off-diagonal entries \(a_{ij}/\sqrt{a_{ii}a_{jj}}\) of \(A\), respectively. Then \(|\det(A)|\leq (d-f)^{n-1}[d+(n-1)f]\), equivalently, \(|\det(A)|\leq (1-f')^{n-1}[1+(n-1)f']\prod_{i=1}^n a_{ii}\), a strengthening of Hadamard's inequality for positive definite matrices. These results are applied to the case the positive definite matrix is obtained as \(AA^T\) from some rectangular matrix \(A\), in particular for the case \(A\) being from one of these classes: entrywise nonnegative, stochastic, \((0,1)\), or \((\pm 1)\). In the latter two cases inequalities of e.g. \textit{H. J. Ryser} [Can. J. Math. 8, 245-249 (1956; Zbl 0071.35903)], \textit{H. Ehlich} [Math. Z. 83, 123-132 (1964; Zbl 0115.24704)], and \textit{C.-S. Cheng} [Ann. Stat. 8, 436-446 (1980; Zbl 0425.62055)] are recovered.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 00B20
Mathematics Subject Classification ID: 15B36
zbMATH DE Number: 1089495
zbMATH Keywords: determinant inequality
zbMATH Keywords: positive definite
zbMATH Keywords: \((0,1)\)-matrices
zbMATH Keywords: \((-1,+1)\)-matrices
zbMATH Keywords: nonnegative matrices
zbMATH Keywords: incidence matrices
zbMATH Keywords: \((n,k,\lambda\)-design
reviewed by: Alexander Kovačec
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1089495