Some monotonicity properties of Schur powers of matrices and related inequalities
type: Item
instance of: scholarly article
title: Some monotonicity properties of Schur powers of matrices and related inequalities
zbMATH Open document ID: 0575.15006
DOI: 10.1016/0024-3795(85)90207-1
author: Samuel Karlin
author: Friedemann Ost
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: Having as a motivation certain probability models, the authors study some inequalities for the spectral radius \(\rho\) of the Schur product of matrices in some conditions. They prove for example: 1. \(\rho\) (A\(\circ B)\leq [\rho (A\circ A)\rho (B\circ B)]^{1/2}\), A and B nonnegative [conditions are given for the equality to occur when A and B are irreducible and when they are stochastic]; 2. If \(\rho_ r=\rho (P^{[r]})\), P nonnegative irreducible, then \(\rho^ 2_ r\leq \rho_{r-1}\rho_{r+1}\) (log \(\rho\) \({}_ r\) convex) for all positive r; 3. For a finite stationary Markov chain, the entropy is given by \((d\rho_ r/dr)_{r=1}\); 4. Inequalities of the form \(\rho (Q^{[\phi]})\leq \rho (P^{[\phi]})\) hold for stochastic matrices and for functions \(\phi\) under given conditions.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 3918572
zbMATH Keywords: entropy
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90207-1
OpenAlex ID: W1967623065
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description: scientific article; zbMATH DE number 3918572