Math 603 - Matrix Analysis

Class: TuTh 4:00 - 5:15PM in Math & Psychology 010 (8/28-12/10/2024), Instructor: Bedrich Sousedik

Topics in this course will include a review of basic matrix operations, determinants, rank, matrix inverse and solving linear equations. The course then will study partitioned matrices, eigenvalues and eigenvectors, spectral decomposition, singular-value decomposition, Jordan canonical form, orthogonal projections, idempotent matrices, quadratic forms, extrema of quadratic forms, non-negative definite and positive definite matrices, and matrix derivatives.

Prerequisite: MATH 221, 251 and 301 or permission of the instructor.

The detailed class syllabus (pdf) is here. However, look below to see the actual progress of the class.

Syllabus of the master's comprehensive exam can be found here.

eigenvectors and eigenvalues explained visually

8/29 Introduction.

9/3 Chapter 1: Linear Equations.

9/5 Chapter 2: Rectangular systems and echelon form. 3.2-3.6 Matrix algebra. Hw 1 (due 9/17).

9/10 3.2-3.6 Matrix algebra. 3.7-3.8 Matrix inversion.

9/12 3.7-3.8 Matrix inversion, inverses of sums. 3.8 Neumann series and sensitivity. HW2 (due 9/24).

9/17 3.9-3.10 Elementary matrices, LU factorization.

9/19 No class.

9/24 4.1 Spaces and subspaces. 4.2 Four fundamental subspaces. Hw 3 (due 10/3).

9/26 4.2 Four fundamental subspaces. 4.2 N(A^T)=R(P_2^T).

10/1 4.2 Four fundamental subspaces. 4.3 Linear independence. Hw 4 (due 10/10).

10/3 4.4 Basis and dimension. 4.5 More about rank.

10/8 4.5 More about rank: products (A^T)A, AA^T, Normal equations. 4.6 Classical least squares.

10/10 4.6 Classical least squares.

10/15 Exam #1. (up to Section 4.5)

10/17 4.7 Linear transformations.

10/22 4.8 Change of basis and similarity.

10/24 4.9 Invariant subspaces. Hw 5 (due 11/5).

10/29 5.1-2 Vector and matrix norms.

10/31 5.3 Inner product spaces. 5.4 Orthogonal vectors. Hw 6 (due 11/14).

11/5 5.5 Gram-Schmidt procedure, in matrix notation, classical and modified variant, QR decomposition, QR and the least squares.

11/7 Talk by Sebastian Deffner “Heat, power, action!” an CNMS Science Discovery Series Event.

11/12 5.6 Unitary and orthogonal matrices. 5.7 Orthogonal reduction.

11/14 5.9 Complementary subspaces.

11/19 5.11 Orthogonal decomposition. 5.13 Orthogonal projection. Hw 7 (due 12/3).

11/21 Exam #2. (Sections 4.7-5.5)

11/26 6.1-6.2 Determinants.

11/28 Thanksgiving (no class).

12/3 7.1 Elementary properties of eigensystems. 7.2 Diagonalization by similarity transformations.

12/5 7.2 Diagonalization by similarity transformations.

12/10

Tuesday 12/17, 3:30-5:30pm Final.