Linear Algebra and Its Applications

Preface


1 Gaussian Elimination

1.1 Introduction

1.2 An Example of Gaussian Elimination

1.3 Matrix Notation and Matrix Multiplication

1.4 Gaussian Elimination = Triangular Factorization

1.5 Row Exchanges, Inverses, and Transposes

1.6 Band Matrices, Applications, and Roundoff

Review Exercises


2 The Theory of Simultaneous Linear Equations

2.1 Vector Spaces and Subspaces

2.2 The Solution of m Equations in η Unknowns

2.3 Linear Independence, Basis, and Dimension

2.4 The Four Fundamental Subspaces

2.5 Orthogonality of Vectors and Subspaces

2.6 Pairs of Subspaces and Products of Matrices

Review Exercises


3 Orthogonal Projections and Least Squares

3.1 Inner Products and Projections onto Lines

3.2 Projections onto Subspaces and Least Squares Approximations

3.3 Orthogonal Bases, Orthogonal Matrices, and Gram-Schmidt Orthogonalization

3.4 The Pseudoinverse and the Singular Value Decomposition

3.5 Weighted Least Squares

Review Exercises


4 Determinants

4.1 Introduction

4.2 The Properties of the Determinant

4.3 Formulas for the Determinant

4.4 Applications of Determinants

Review Exercises


5 Eigenvalues and Eigenvectors

5.1 Introduction

5.2 The Diagonal Form of a Matrix

5.3 Difference Equations and the Powers Ak

5.4 Differential Equations and the Exponential eAT

5.5 The Complex Case: Hermitian and Unitary Matrices

5.6 Similarity Transformations and Triangular Forms

Review Exercises


6 Positive Definite Matrices

6.1 Minima, Maxima, and Saddle Points

6.2 Tests for Positive Definiteness

6.3 Semidefinite and Indefinite Matrices; Ax = λBx

6.4 Minimum Principles and Rayleigh's Quotient

6.5 The Rayleigh- Ritz Principle and Finite Elements


7 Computations with Matrices

7.1 Introduction

7.2 The Norm and Condition Number of a Matrix

7.3 The Computation of Eigenvalues

7.4 Iterative Methods for Ax = b


8 Linear Programming and Game Theory

8.1 Linear Inequalities

8.2 The Simplex Method

8.3 The Theory of Duality

8.4 Network Models

8.5 Game Theory and the Minimax Theorem

Appendix A Linear Transformations, Matrices, and Change of Basis

Appendix Β The Jordan Form

Appendix C Computer Codes for Linear Algebra

References

Solutions to Exercises

Index