A Survey of Matrix Theory and Matrix Inequalities

by Marvin Marcus, Henryk Minc

Overview

Concise yet comprehensive survey covers broad range of topics: convexity and matrices, localization of characteristic roots, proofs of classical theorems and results in contemporary research literature, much more. Undergraduate-level. 1969 edition. Bibliography.

Product Details

ISBN-13:	9780486671024
Publisher:	Dover Publications
Publication date:	09/16/2010
Series:	Dover Books on Mathematics Series
Edition description:	Reprint
Pages:	208
Product dimensions:	5.50(w) x 8.50(h) x (d)

Table of Contents

I. SURVEY OF MATRIX THEORY
1. INTRODUCTORY CONCEPTS
Matrices and vectors.
Matrix operations.
Inverse.
Matrix and vector operations.
Examples.
Transpose.
Direct sum and block multiplication.
Examples.
Kronecker product.
Example.
2. NUMBERS ASSOCIATED WITH MATRICES
Notation.
Submatrices.
Permutations.
Determinants.
The quadratic relations among subdeterminants.
Examples.
Compound matrices.
Symmetric functions; trace.
Permanents.
Example.
Properties of permanents.
Induced matrices.
Characteristic polynomial.
Examples.
Characteristic roots.
Examples.
Rank.
Linear combinations.
Example.
Linear dependence; dimension.
Example.
3. LINEAR EQUATIONS AND CANONICAL FORMS
Introduction and notation.
Elementary operations.
Example.
Elementary matrices.
Example.
Hermite normal form.
Example.
Use of the Hermite normal form in solving Ax = b.
Example.
Elementary column operations and matrices.
Examples.
Characteristic vectors.
Examples.
Conventions for polynomial and integral matrices.
Determinantal divisors.
Examples.
Equivalence.
Example.
Invariant factors.
Elementary divisors.
Examples.
Smith normal form.
Example.
Similarity.
Examples.
Elementary divisors and similarity.
Example.
Minimal polynomial.
Companion matrix.
Examples.
Irreducibility.
Similarity to a diagonal matrix.
Examples.
4. "SPECIAL CLASSES OF MATRICES, COMMUTATIVITY"
Bilinear functional.
Examples.
Inner product.
Example.
Orthogonality.
Example.
Normal matrices.
Examples.
Circulant.
Unitary similarity.
Example.
Positive definite matrices.
Example.
Functions of normal matrices.
Examples.
Exponential of a matrix.
Functions of an arbitrary matrix.
Example.
Representation of a matrix as a function of other matrices.
Examples.
Simultaneous reduction of commuting matrices.
Commutativity.
Example.
Quasi-commutativity.
Example.
Property L.
Examples.
Miscellaneous results on commutativity.
5. CONGRUENCE
Definitions.
Triple diagonal form.
Congruence and elementary operations.
Example.
Relationship to quadratic forms.
Example.
Congruence properties.
Hermitian congruence.
Example.
Triangular product representation.
Example.
Conjunctive reduction of skew-hermitian matrices.
Conjunctive reduction of two hermitian matrices.
II. CONVEXITY AND MATRICES
1. CONVEX SETS
Definitions.
Examples.
Intersection property.
Examples.
Convex polyhedrons.
Example.
Birkhoff theorem.
Simplex.
Examples.
Dimension.
Example.
Linear functionals.
Example.
2. CONVEX FUNCTIONS
Definitions.
Examples.
Properties of convex functions.
Examples.
3. CLASSICAL INEQUALITIES
Power means.
Symmetric functions.
Hölder inequality.
Minkowski inequality.
Other inequalities.
Example.
4. CONVEX FUNCTIONS AND MATRIX INEQUALITIES
Convex functions of matrices.
Inequalities of H. Weyl.
Kantorovich inequality.
More inequalities.
Hadamard product.
5. NONNEGATIVE MATRICES
Introduction.
Indecomposable matrices.
Examples.
Fully indecomposable matrices.
Perron-Frobenius theorem.
Example.
Nonnegative matrices.
Examples.
Primitive matrices.
Example.
Doubly stochastic matrices.
Examples.
Stochastic matrices.
III. LOCALIZATION OF CHARACTERISTIC ROOTS
1. BOUNDS FOR CHARACTERISTIC ROOTS
Introduction.
Bendixson's theorems.
Hirsch's theorems.
Schur's inequality (1909).
Browne's theorem.
Perron's theorem.
Schneider's theorem.
2. REGIONS CONTAINING CHARACTERISTIC ROOTS OF A GENERAL MATRIX.
Lévy-Desplanques theorem.
Gersgorin discs.
&

Editorial Reviews

A reprint of the second, corrected printing (1969) of the work for advanced undergraduates first published by Prindle, Weber & Schmidt in 1964. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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