A Survey of Matrix Theory and Matrix Inequalities

Marvin Marcus, Henryk Minc

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486671024
• eBook ISBN: 9780486153063
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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.

Survey of Matrix Theory

I

1 Introductory Concepts

1.1 Matrices and vectors

In this book we will be dealing with the following sets:

(i) The field of real numbers R.

(ii) The field of complex numbers C.

(iii) The ring of all integers (positive, negative, and zero) I.

(iv) The set of all nonnegative real numbers P.

(v) The ring of polynomials in some indeterminate (“variable”) λ with real (complex) coefficients R[λ] (C[λ]).

Let F be any one of these sets and suppose that a₁₁, a₁₂, a₁₃, ···, a_mn is a collection of mn elements in F. The rectangular array of these elements

consisting of m rows and n columns is called an m × n matrix. We write A = (a_ij). A more formal but less suggestive definition is this: a matrix A is a function on the set of pairs of integers (i,j), 1 ≤ i ≤ m, 1 ≤ j ≤ n, with values in F in which a_ij designates the value of A at the pair (i,j). The array 1.1(1) is then just a convenient way of exhibiting the range of the function A. The element a_ij is called the (i,j) entry of A. The ith row of A is the sequence a_i1, a_i2, ···, a_in, and the jth column is the sequence a_1j, a_2j, ···, a_mj. Since the a_ij are in F, we say that A is an m × n matrix over F. The totality of such matrices will be designated by M_m,n(F). In case n = m, we say that A is an n-square matrix. Denote by M_n(F) the set of all n-square matrices over F. A row (column) vector over F is just an element of M_1,n(F) (M_m,1(F)). If A ∈ M_m,n(F), then A_(i) = (a_i1, ···, a_in ∈ M_1,n(F), (i = 1, ···, m), designates the ith row vector of A. Similarly, the column vector A⁽ⁱ⁾ ∈ M_m,1(F) designates the m × 1 matrix whose (i,j) entry is a_ij, (i = 1, ···, m). The (i,j) entry of A is sometimes designated by A_ij also. In general, a k-vector v over F is just an ordered k-tuple of elements of F, (a₁, a₂, ···, a_k); a_i...

Table of contents

• Cover
• Title Page
• Dedication
• Preface
• Contents
• I. Survey of Matrix Theory
• II. Convexity And Matrices
• III. Localization of Characteristic Roots
• Index