A Survey of Matrix Theory and Matrix Inequalities, Volume 14

Front Cover
Courier Corporation, 1992 - Mathematics - 180 pages
0 Reviews

Written for advanced undergraduate students, this highly regarded book presents an enormous amount of information in a concise and accessible format. Beginning with the assumption that the reader has never seen a matrix before, the authors go on to provide a survey of a substantial part of the field, including many areas of modern research interest.
Part One of the book covers not only the standard ideas of matrix theory, but ones, as the authors state, "that reflect our own prejudices," among them Kronecker products, compound and induced matrices, quadratic relations, permanents, incidence matrices and generalizations of commutativity.
Part Two begins with a survey of elementary properties of convex sets and polyhedra and presents a proof of the Birkhoff theorem on doubly stochastic matrices. This is followed by a discussion of the properties of convex functions and a list of classical inequalities. This material is then combined to yield many of the interesting matrix inequalities of Weyl, Fan, Kantorovich and others. The treatment is along the lines developed by these authors and their successors and many of their proofs are included. This chapter contains an account of the classical Perron Frobenius-Wielandt theory of indecomposable nonnegative matrices and ends with some important results on stochastic matrices.
Part Three is concerned with a variety of results on the localization of the characteristic roots of a matrix in terms of simple functions of its entries or of entries of a related matrix. The presentation is essentially in historical order, and out of the vast number of results in this field the authors have culled those that seemed most interesting or useful. Readers will find many of the proofs of classical theorems and a substantial number of proofs of results in contemporary research literature.

 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Selected pages

Contents

Survey of Matrix Theory
1
Numbers Associated with Matrices
9
Linear Equations and Canonical Forms
30
Special Classes of Matrices Commutativity
59
Congruence
81
Convexity and Matrices
93
Convex Functions
101
Classical Inequalities
105
Convex Functions and Matrix Inequalities
112
Nonnegative Matrices
121
Localization of Characteristic Roots
139
Regions Containing Characteristic Roots of a General Matrix
145
Characteristic Roots of Special Types of Matrices
152
The Spread of a Matrix
167
The Field of Values of a Matrix
168
Copyright

Other editions - View all

Common terms and phrases

References to this book

All Book Search results »

Bibliographic information