Chapter 0 Review and miscellanea

0.0 Introduction

0.1 Vector spaces

0.2 Matrices

0.3 Determinants

0.4 Rank

0.5 Nonsingularity

0.6 The usual inner product

0.7 Partitioned matrices

0.8 Determinants again

0.9 Special types of matrices

0.10 Change of basis

Chapter 1 Eigenvalues,eigenvectors,and similarity

1.0 Introduction

1.1 The eigenvalue-eigenvector equation

1.2 The characteristic polynomial

1.3 Similarity

1.4 Eigenvectors

Chapter 2 Unitary equivalence and normal matrices

2.0 Introduction

2.1 Unitary matrices

2.2 Unitary equivalence

2.3 Schur's unitary triangularization theorem

2.4 Some implications of Schur's theorem

2.5 Normal matrices

2.6 QR factorization and algorithm

Chapter 3 Canonical forms

3.0 Introduction

3.1 The Jordan canonical form:a proof

3.2 The Jordan canonical form:some observations and applications

3.3 Polynomials and matrices:the minimal polynomial

3.4 Other canonical forms and factorizations

3.5 Triangular factorizations

Chapter 4 Hermitian and symmetric matrices

4.0 Introduction

4.1 Definitions,properties,and characterizations of Hermitian matrices

4.2 Variational characterizations of eigenvalues of Hermitian matrices

4.3 Some applications of the variational characterizations

4.4 Complex symmetric matrices

4.5 Congruence and simultaneous diagonalization of Hermitian and symmetric matrices

4.6 Consimilarity and condiagonalization

Chapter 5 Norms for vectors and matrices

5.0 Introduction

5.1 Defining properties of vector norms and inner products

5.2 Examples of vector norms

5.3 Algebraic properties of vector norms

5.4 Analytic properties of vector norms

5.5 Geometric properties of vector norms

5.6 Matrix norms

5.7 Vector norms on matrices

5.8 Errors in inverses and solutions of linear systems

Chapter 6 Location and perturbation of eigenvalues

6.0 Introduction

6.1 Ger？gorin discs

6.2 Ger？gorin discs-a closer look

6.3 Perturbation theorems

6.4 Other inclusion regions

Chapter 7 Positive definite matrices

7.0 Introduction

7.1 Definitions and properties

7.2 Characterizations

7.3 The polar form and the singular value decomposition

7.4 Examples and applications of the singular value decomposition

7.5 The Schur product theorem

7.6 Congruence:products and simultaneous diagonalization

7.7 The positive semidefinite ordering

7.8 Inequalities for positive definite matrices

Chapter 8 Nonnegative matrices

8.0 Introduction

8.1 Nonnegative matrices-inequalities and generalities

8.2 Positive matrices

8.3 Nonnegative matrices

8.4 Irreducible nonnegative matrices

8.5 Primitive matrices

8.6 A general limit theorem

8.7 Stochastic and doubly stochastic matrices

Appendices

A Complex numbers

B Convex sets and functions

C The fundamental theorem of algebra

D Continuous dependence of the zeroes of a polynomial on its coefficients

E Weierstrass's theorem

References

Notation

Index