Open Access Research

Sum of squared logarithms - an inequality relating positive definite matrices and their matrix logarithm

Mircea Bîrsan12, Patrizio Neff1* and Johannes Lankeit1

Author Affiliations

1 Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany

2 Department of Mathematics, University ‘A.I. Cuza’ of Iaşi, Iaşi, Romania

For all author emails, please log on.

Journal of Inequalities and Applications 2013, 2013:168  doi:10.1186/1029-242X-2013-168

Published: 12 April 2013

Abstract

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M1">View MathML</a> be such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M2">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M3">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M4">View MathML</a>

This can also be stated in terms of real positive definite <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M5">View MathML</a>-matrices <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M6">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M7">View MathML</a>: If their determinants are equal, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M8">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M9">View MathML</a>

where log is the principal matrix logarithm and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M10">View MathML</a> denotes the Frobenius matrix norm. Applications in matrix analysis and nonlinear elasticity are indicated.

MSC: 26D05, 26D07.

Keywords:
matrix logarithm; elementary symmetric polynomials; inequality; characteristic polynomial; positive definite matrices; means