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A Survey of Matrix Theory and Matrix Inequalities

Marvin Marcus and Henryk Minc
Publisher: 
Dover Publications
Publication Date: 
1992
Number of Pages: 
180
Format: 
Paperback
Price: 
12.95
ISBN: 
9780486671024
Category: 
Monograph
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mehdi Hassani
, on
10/6/2011
]

Considering a collection of numbers (or other mathematical objects) as a rectangular array is an idea used by many people from various branches of mathematics, statistics, computer sciences, and engineering. The properties of such arrays, which we call matrices, are also at the base of recent developments in combinatorics, coding theory, and many other subjects.

As its title indicates, the book under review is a concise account of the theory of matrices, giving a quick but powerful picture of the theory focusing mostly on inequalities. The book has three main parts. The first part gives a survey of matrix theory. It contains various lists of results, many of them with no proof. As a foundation for the next two parts, this survey includes some sections introducing numbers related to matrices, such as the determinant, permanent, trace, and characteristic roots (aka eigenvalues).

The second part, which is indeed the main part and heart of the book, contains various inequalities involving the above numbers for various kinds of matrices. One may find a list of them in the table of contents. Convexity plays an important role: it is the first topic discussed, and the two first sections study it. In the process, the authors give a quick review of important classical inequalities, which seems very nice for the people working on inequality theory. The third part of the book studies the location of characteristic roots by giving several inequalities involving them. 

The book includes compactly presented information on many classical theorems, relations, and inequalities for matrices. This makes the book a friendly reference for researchers who deal with matrices, especially those who work with inequalities.

The book has no exercises, so is it not a textbook. But the structure of the chapters and the method of presentation are much like what one would find in a text. I believe that if an instructor considers some of the facts that are merely listed as exercises, then the book could be used as a textbook in an advanced course for undergraduates, or even as a part of some graduate courses.

I highly recommend this book for anyone involved in a course related to or using matrix theory: there is no doubt that they will find some very nice and useful information.


Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His field of interest is Elementary, Analytic and Probabilistic Number Theory.  

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.
Circu
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 matr
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.

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