A Fresh Variational‐Analysis Look at the Positive Semidefinite Matrices World

A Fresh Variational‐Analysis Look at the Positive Semidefinite Matrices World

0.00 Avg rating0 Votes
Article ID: iaor20123942
Volume: 153
Issue: 3
Start Page Number: 551
End Page Number: 577
Publication Date: Jun 2012
Journal: Journal of Optimization Theory and Applications
Authors: ,
Keywords: matrices
Abstract:

Engineering sciences and applications of mathematics show unambiguously that positive semidefiniteness of matrices is the most important generalization of non‐negative real numbers. This notion of non‐negativity for matrices has been well‐studied in the literature; it has been the subject of review papers and entire chapters of books. This paper reviews some of the nice, useful properties of positive (semi)definite matrices, and insists in particular on (i) characterizations of positive (semi)definiteness and (ii) the geometrical properties of the set of positive semidefinite matrices. Some properties that turn out to be less well‐known have here a special treatment. The use of these properties in optimization, as well as various references to applications, is spread all the way through. The ‘raison d’être’ of this paper is essentially pedagogical; it adopts the viewpoint of variational analysis, shedding new light on the topic. Important, fruitful, and subtle, the positive semidefinite world is a good place to start with this domain of applied mathematics.

Reviews

Required fields are marked *. Your email address will not be published.