Special Matrices and Their Applications in Numerical Mathematics

Second Edition

Miroslav Fiedler

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486466750
• eBook ISBN: 9780486783482
• Edition: 2
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

This revised and corrected second edition of a classic book on special matrices provides researchers in numerical linear algebra and students of general computational mathematics with an essential reference.
Author Miroslav Fiedler, a Professor at the Institute of Computer Science of the Academy of Sciences of the Czech Republic, Prague, begins with definitions of basic concepts of the theory of matrices and fundamental theorems. In subsequent chapters, he explores symmetric and Hermitian matrices, the mutual connections between graphs and matrices, and the theory of entrywise nonnegative matrices. After introducing M-matrices, or matrices of class K, Professor Fiedler discusses important properties of tensor products of matrices and compound matrices and describes the matricial representation of polynomials. He further defines band matrices and norms of vectors and matrices. The final five chapters treat selected numerical methods for solving problems from the field of linear algebra, using the concepts and results explained in the preceding chapters.

1

Basic concepts of matrix theory

This introductory chapter is essentially a survey of well known definitions and general principles from matrix theory which will be useful in the sequel.

1.1 Matrices

A matrix of type m-by-n or, equivalently, an m × n matrix, is a rectangular array of ran numbers (usually real or complex) arranged in m rows and n columns (m, n positive integers):

We call the number a_lk the entry of the matrix in the ith row and the kth. column. The set of m × n matrices with real entries will be denoted by R^m×n. Analogously, the set of m × n matrices with complex entries will be denoted by C^m×n. In some cases, entries can be polynomials, variables, functions, etc.

In this terminology, matrices with only one column (thus, n = 1) will be called column vectors and matrices with only one row (thus, m = 1) row vectors. In such case, we write R^m instead of R^m×1 and vectors will always be column vectors, unless we specify otherwise.

It is advantageous to denote the matrix by a single symbol, say, A, or [a_lk], C, etc.

Two matrices A = [a_ik], B = [b_ik] are equal, written A = B, if and only if they are both m × n matrices for some positive integers m and n, and a_ik = b_ik for i = 1, …, m, j = 1, …, n.

The importance of matrices is that one can introduce operations generalizing operations with numbers.

Matrices of the same type can be added: If A = [a_ik], B = [b_ik] then A + B is the matrix [a_ik + b_ik]. The operation of addition is thus entrywise. We admit also multiplication of a matrix by a number (real, complex, a parameter, etc.): If A = [a_ik] and if α is a number (also called scalar), then αA is the matrix [αa_ik] of the same type as A.

An m × n matrix A = [a_ik] can be multiplied by an n × p matrix B = [b_kl] as follows: it is the m × p matrix C = [c_il], where

It is important to notice that the matrices A and B can be multiplied (in this order) only if the number of columns of A is the same as the number of rows in B. Also, the entries of A and B should be multiplicable. In general, the product AB is not equal to BA, even if the multiplication of both products is possible. On the other hand, the mult...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Preface
• 1 Basic Concepts of Matrix Theory
• 2 Symmetric Matrices. Positive Definite and Semidefinite Matrices
• 3 Graphs and Matrices
• 4 Nonnegative Matrices. Stochastic and Doubly Stochastic Matrices
• 5 M-Matrices (Matrices of Classes K and K0)
• 6 Tensor Product of Matrices. Compound Matrices
• 7 Matrices and Polynomials. Stable Matrices
• 8 Band Matrices
• 9 Norms and Their Use for Estimation of Eigenvalues
• 10 Direct Methods for Solving Linear Systems
• 11 Iterative Methods for Solving Linear Systems
• 12 Matrix Inversion
• 13 Numerical Methods for Computing Eigenvalues of Matrices
• 14 Sparse Matrices
• Bibliography
• Index