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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

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

Library of Congress subject headings for this publication: Matrices