Not only is matrix theory significant in a wide range of fields mathematical economics, quantum physics, geophysics, electrical network synthesis, crystallography, and structural engineering, among others-but with the vast proliferation of digital computers, knowledge of matrix theory is a must for every modern engineer, mathematician, and scientist. Matrices represent linear transformations from a finiteset of numbers to another finite set of numbers.
Since many important problems are linear, and since digital computers with finite memory manipulate only finite sets of numbers, the solution of linear problems by digital computers usually involves matrices. Developed from the author's course on matrix theory at the California
Institute of Technology, the book begins with a concise presentation of the theory of determinants, continues with a discussion of classical linear algebra, and an optional chapter on the use of matrices to solve systems of linear triangularizations of Hermitian and nonHermitian matrices, as well as a chapter presenting a proof of the difficult and important matrix theory of Jordan. The book concludes with discussions of variational principles and perturbation theory of matrices, matrix numerical analysis, and an introduction to the subject of linear computations.
The book is designed to meet many different needs, and because it is mathematically rigorous, it may be used by students of pure and applied mathematics. Since it is oriented towards applications, it is valuable to students of engineering, science, and the social sciences. And because it contains the basic preparation in matrix theory required for numerical analysis, it can be used by students whose main interest is computers. The book assumes very little mathematical preparation, and except for the single section on the continuous dependence of eigenvalues on matrices, a knowledge of elementary algebra and calculus is sufficient.
1. Determinants 1.1 Introduction 1.2 The Definition of a Determinant 1.3 Properties of Determinants 1.4 Row and Column Expansions 1.5 Vectors and Matrices 1.6 The Inverse Matrix 1.7 The Determinant of a Matrix Product 1.8 The Derivative of a Determinant 2. The Theory of Linear Equations 2.1 Introduction 2.2 Linear Vector Spaces 2.3 Basis and Dimension 2.4 Solvability of Homogeneous Equations 2.5 Evaluation of Rank by Determinants 2.6 The General m x n Inhomogeneous System 2.7 Least-Squares Solution of Unsolvable Systems 3. Matrix Analysis of Differential Equations 3.1 Introduction 3.2 Systems of Linear Differential Equations 3.3 Reduction to the Homogeneous System 3.4 Solution by the Exponential Matrix 3.5 Solution by Eigenvalues and Eigenvectors 4. Eigenvalues, Eigenvectors, and Canonical Forms 4.1 Matrices with Distinct Eigenvalues 4.2 The Canonical Diagonal Form 4.3 The Trace and Other Invariants 4.4 Unitary Matrices 4.5 The Gram-Schmidt Orthogonalization Process 4.6 Principal Axes of Ellipsoids 4.7 Hermitian Matrices 4.8 Mass-spring Systems; Positive Definiteness; Simultaneous Diagonalization 4.9 Unitary Triangularization 4.10 Normal Matrices 5. The Jordan Canonical Form 5.1 Introduction 5.2 Principal Vectors 5.3 Proof of Jordan's Theorem 6. Variational Principles and Perturbation Theory 6.1 Introduction 6.2 The Rayleigh Principle 6.3 The Courant Minimax Theorem 6.4 The Inclusion Principle 6.5 A Determinant-criterion for Positive Definiteness 6.6 Determinants as Volumes; Hadamard's Inequality 6.7 Weyl's Inequalities 6.8 Gershgorin's Theorem 6.9 Vector Norms and the Related Matrix Norms 6.10 The Condition-Number of a Matrix 6.11 Positive and Irreducible Matrices 6.12 Perturbations of the Spectrum 6.13 Continuous Dependence of Eigenvalues on Matrices 7. Numerical Methods 7.1 Introduction 7.2 The Method of Elimination 7.3 Factorization by Triangular Matrices 7.4 Direct Solution of Large systems of Linear Equations 7.5 Reduction of Rounding Error 7.6 The Gauss-Seidel and Other Iterative Methods 7.7 Computation of Eigenvectors from Known Eigenvalues 7.8 Numerical Instability of the Jordan Canonical Form 7.9 The Method of Iteration for Dominant Eigenvalues 7.10 Reduction to Obtain the Smaller Eigenvalues 7.11 Eigenvalues and Eigenvectors of Tridiagonal and Hessenberg Matrices 7.12 The Method of Householder and Bauer 7.13 Numerical Identification of Stable Matrices 7.14 Accurate Unitary Reduction to Triangular Form 7.15 The QR Method for Computing Eigenvalues Index
