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1999 | Buch

Elementary Linear Algebra Review Kapitel lesen Erstes Kapitel lesen

Matrix Theory

Basic Results and Techniques

verfasst von: Fuzhen Zhang

Verlag: Springer New York

Buchreihe : Universitext

Enthalten in: Professional Book Archive

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SUCHEN

Über dieses Buch

The aim of this book is to concisely present fundamental ideas, results, and techniques in linear algebra and mainly matrix theory. The book contains eight chapters covering various topics ranging from similarity and special types of matrices to Schur complements and matrix normality. Each chapter focuses on the results, techniques, and methods that are beautiful, interesting, and representative, followed by carefully selected problems. For many theorems several different proofs are given. The book can be used as a text or a supplement for a linear algebra and matrix theory class or seminar for senior or graduate students. The only prerequisites are a decent background in elementary linear algebra and calculus. The book can also serve as a reference for instructors and researchers in the fields of algebra, matrix analysis, operator theory, statistics, computer science, engineering, operations research, economics, and other fields.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Elementary Linear Algebra Review
Abstract
We briefly review, mostly without proof, the basic concepts and results taught in an elementary linear algebra course. The subjects are vector spaces, basis and dimension, linear transformations, the similarity of matrices, and inner product spaces.
Fuzhen Zhang
Chapter 2. Partitioned Matrices
Abstract
This chapter is devoted to the techniques of partitioned (block) matrices. Topics include elementary operations, determinants, and inverses of partitioned matrices. We begin with the elementary operations of block matrices, followed by discussions of the inverse and rank of the sum and product of matrices. We then present four different proofs of the theorem that the products AB and BA of matrices A and B of sizes m × n and n × m, respectively, have the same nonzero eigenvalues. At the end of this chapter we discuss the often-used matrix technique of continuity argument.
Fuzhen Zhang
Chapter 3. Matrix Polynomials and Canonical Forms
Abstract
This chapter is devoted to matrix decompositions. The main studies are on the Schur decomposition, spectral decomposition, singular value decomposition, Jordan decomposition, and numerical range. Attention is also paid to the polynomials that annihilate matrices, especially the minimal and characteristic polynomials, and to the similarity of a complex matrix to a real matrix.
Fuzhen Zhang
Chapter 4. Special Types of Matrices
Abstract
This chapter studies special types of matrices. They are: idempotent matrices, nilpotent matrices, involutary matrices, projection matrices, tridiagonal matrices, circulant matrices, Vandermonde matrices, Hadamard matrices, permutation matrices, and doubly stochastic matrices. These matrices are often used in many subjects of mathematics and in other fields.
Fuzhen Zhang
Chapter 5. Unitary Matrices and Contractions
Abstract
This chapter studies unitary matrices and contractions. Section 5.1 gives basic properties of unitary matrices, Section 5.2 discusses the structure of real orthogonal matrices under similarity, and Section 5.3 develops metric spaces and the fixed-point theorem of strict contractions. Section 5.4 deals with the connections of contractions with unitary matrices, Section 5.5 concerns the unitary similarity of real matrices, and Section 5.6 presents a trace inequality for unitary matrices, relating the average of the eigenvalues of each of two unitary matrices to that of their product.
Fuzhen Zhang
Chapter 6. Positive Semidefinite Matrices
Abstract
This chapter studies the positive semidefinite matrices, concentrating primarily on the inequalities of this type of matrix. The main goal is to present the fundamental results and show some often-used techniques. Section 6.1 gives the basic properties, Section 6.2 treats the Löwner partial ordering of positive semidefinite matrices, and Section 6.3 presents some inequalities of principal submatrices. Section 6.4 derives inequalities of partitioned positive semidefinite matrices using Schur complements, while Sections 6.5 and 6.6 investigate the Hadamard product and Kronecker product and related matrix inequalities. Finally, Section 6.7 shows matrix inequalities of the Cauchy-Schwarz type.
Fuzhen Zhang
Chapter 7. Hermitian Matrices
Abstract
This chapter contains fundamental results of Hermitian matrices and demonstrates the basic techniques used to derive the results. Section 7.1 presents equivalent conditions to matrix Hermitity, Section 7.2 gives some trace inequalities and discusses a necessary and sufficient condition for a square matrix to be a product of two Hermitian matrices, and Section 7.3 develops the min-max theorem and the interlacing theorem for eigenvalues. Section 7.4 deals with the eigenvalue and singular value inequalities for the sum of Hermitian matrices, and Section 7.5 shows a matrix triangle inequality.
Fuzhen Zhang
Chapter 8. Normal Matrices
Abstract
A great deal of elegant work has been done for normal matrices. The goal of this chapter is to present basic results and methods on normal matrices. Section 8.1 gives conditions equivalent to the normality of matrices, Section 8.2 focuses on a special type of normal matrix with entries consisting of zeros and ones, Section 8.3 studies the positive semidefinite matrix (A*A)½ associated with a matrix A, and finally Section 8.4 shows majorization inequalities that, when equality holds, result in the normality of the matrix.
Fuzhen Zhang
Backmatter
Metadaten
Titel
Matrix Theory
verfasst von
Fuzhen Zhang
Copyright-Jahr
1999
Verlag
Springer New York
Electronic ISBN
978-1-4757-5797-2
Print ISBN
978-1-4757-5799-6
DOI
https://doi.org/10.1007/978-1-4757-5797-2