Matrix Methods By University Of Minnesota Course Syllabus

Overview: This course provides a rigorous introduction to matrix methods essential for modern computational science and machine learning. Designed for learners with a solid foundation in linear algebra, it covers advanced matrix factorizations, singular value decomposition, eigenvalue analysis, and modern applications in data science and numerical algorithms. The course spans approximately 16-20 weeks of part-time study, with an estimated 8-10 hours per week. Each module combines theoretical depth with practical implementation in MATLAB or Python, preparing learners for research and industry applications.

Module 1: Matrix Factorizations

Estimated time: 35 hours

• LU decomposition with partial and complete pivoting
• QR decomposition via Gram-Schmidt and Householder methods
• Cholesky decomposition for symmetric positive definite matrices
• Applications to solving linear systems and matrix inversion
• Forward and backward substitution algorithms

Module 2: Singular Value Decomposition

Estimated time: 45 hours

• Theory and derivation of the singular value decomposition (SVD)
• Low-rank matrix approximation and Eckart-Young theorem
• Computation of pseudoinverses and their role in least squares
• Data compression using truncated SVD
• Applications in image processing and dimensionality reduction

Module 3: Least Squares and Regularization

Estimated time: 30 hours

• Formulation of overdetermined and underdetermined systems
• Normal equations and their numerical instability
• Regularization techniques: Tikhonov and ridge regression
• SVD-based solutions to ill-conditioned problems
• Implementation in MATLAB/Python with real datasets

Module 4: Eigenvalue Methods

Estimated time: 35 hours

• Power iteration and inverse iteration algorithms
• QR algorithm for eigenvalue computation
• Spectral theorem and diagonalization of symmetric matrices
• Positive definite matrices and their properties
• Applications to dynamical systems and stability analysis

Module 5: Special Topics in Numerical Linear Algebra

Estimated time: 25 hours

• Sparse matrix representations and algorithms
• Randomized methods for low-rank approximation
• Matrix functions: exponentials, logarithms, and their uses
• Case studies in machine learning: PCA, recommender systems

Module 6: Final Project

Estimated time: 20 hours

• Choose a real-world dataset or scientific problem
• Apply SVD, QR, or eigenvalue methods to extract insights
• Submit code, analysis, and a short technical report

Prerequisites

• Strong foundation in linear algebra (vectors, matrices, rank, determinants)
• Familiarity with basic numerical methods and computational error
• Programming experience in MATLAB or Python

What You'll Be Able to Do After

• Master singular value decomposition and its applications in data science
• Implement and compare advanced matrix factorizations (LU, QR, Cholesky)
• Solve least squares problems with regularization and numerical stability
• Apply eigenvalue methods to analyze dynamical systems and convergence
• Develop computational linear algebra skills in MATLAB/Python