Carl Meyer

MA 523 - Matrix Analysis and Applied Linear Algebra

Course Outline


1. Linear Equations (Review)

1.1 Introduction

1.2 Gaussian Elimination And Matrices

1.3 Gauss-Jordan Method

1.4 Two-Point Boundary Value Problems

1.5 Making Gaussian Elimination Work

1.6 Ill-Conditioned Systems

2. Echelon Forms (Review)

2.1 Row Echelon Form And Rank

2.2 The Reduced Row Echelon Form

2.3 Consistency Of Linear Systems

2.4 Homogeneous Systems

2.5 Nonhomogeneous Systems

3. Matrix Algebra (Review)

3.1 From ancient China to Arthur Cayley

3.2 Addition, Scalar Multiplication, And Transposition

3.3 Linearity

3.5 Matrix Multiplication

3.6 Properties Of Matrix Multiplication

3.7 Matrix Inversion

3.8 Inverses Of Sums and Sensitivity

3.9 Elementary Matrices And Equivalence

3.10 The LU Factorization

4. Vector Spaces

4.1 Spaces And Subspaces

4.2 Four Fundamental Subspaces

4.3 Linear Independence

4.4 Basis And Dimension

4.5 More About Rank

4.6 Classical Least Squares

4.7 Linear Transformations

4.8 Change Of Basis And Similarity

4.9 Invariant Subspaces

5. Norms, Inner Products, and Orthogonality

5.1 Vector Norms

5.2 Matrix Norms

5.3 Inner Product Spaces

5.4 Orthogonal Vectors

5.5 Gram-Schmidt Procedure

5.6 Unitary and Orthogonal Matrices

5.7 Orthogonal Reduction

5.8 Discrete Fourier Transform

5.9 Complementary Subspaces

5.10 Range-Nullspace Decomposition

5.11 Orthogonal Decomposition

5.12 Singular Value Decomposition

5.13 Orthogonal Projection

5.14 Why Least Squares?

5.15 Angles Between Subspaces

6. Determinants (Review)

6.1 Determinants

6.2 Additional Properties Of Determinants

7. Eigenvalues And Eigenvectors (As time permits)

7.1 Elementary Properties Of Eigensystems

7.2 Diagonalization by Similarity Transformations

7.3 Functions Of Diagonalizable Matrices

7.4 Systems Of Differential Equations

7.5 Normal Matrices

7.6 Positive Definite Matrices

7.7 Nilpotent Matrices And Jordan Structure

7.8 Jordan Form

7.9 Functions of Nondiagonalizable Matrices

7.10 Difference Equations, Limits, Summability

7.11 Minimum Polynomials and Krylov Methods