List of Topics

The following is a list of topics that will be covered (time permitting) in this course along with the chapter and section numbers. The chapter and section numbers are taken from the textbook "Matrix Analysis", by Roger A. Horn and Charles R. Johnson. There will also be several concepts presented that are not in the book.

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

2.1 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 FORM

3.1 The Jordan canonical form: a proof
3.2 The Jordan canonical form
3.3 Polynomials and matrices

CHAPTER 4. HERMITIAN AND SYMMETRIC MATRICES

4.1 Definitions, properties, and characterizations
4.2 Variational characterizations of eigenvalues
4.3 Some applications

CHAPTER 5. NORMS FOR VECTORS AND MATRICES

5.1 Properties of vectors norms and inner products
5.2 Examples of vector norms
5.6 Matrix norms

CHAPTER 6. LOCATION AND PERTURBATION OF EIGENVALUES

6.1 Gersgorin discs

CHAPTER 7. POSITIVE DEFINITE MATRICES

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