Course Description

MA 523 develops the matrix-theoretic foundations behind modern numerical linear algebra. This is a well establish course by Prof. Ilse Ipsen that focuses on vector spaces, linear transformations, orthogonality, matrix decompositions, least squares, singular values, and the behavior of matrix computations in finite precision.

Topics Covered

• Vector spaces, linear transformations, and matrix representations
• Orthogonality, rotations, reflections, and projectors
• Matrix norms, conditioning, and floating-point arithmetic
• LU, Cholesky, QR, and least-squares problems
• Generalized inverses and definite matrices
• Singular values, SVD, low-rank approximation, and PCA
• MATLAB for matrix computation and numerical experimentation

Learning Outcomes

By the end of the course, students will be able to:

• Work fluently with vectors, matrices, matrix products, and linear transformations
• Solve linear systems and least-squares problems using matrix factorizations
• Use SVD for rank, approximation, conditioning, and data analysis
• Understand how roundoff error affects matrix computations
• Use MATLAB to implement and test matrix algorithms

Textbooks & Resources

The course uses online texts available through NC State Library and Moodle:

1. Numerical Matrix Analysis by I. C. F. Ipsen

2. Numerical Methods in Matrix Computations by Åke Björck

3. Matrix Analysis, 2nd ed. by Roger A. Horn and Charles R. Johnson

4. Fundamentals of Matrix Computations by David S. Watkins

5. Applied Linear Algebra by P. J. Olver and C. Shakiban

Helpful Tools

• MATLAB Onramp — recommended for MATLAB basics
• MathWorks MATLAB Documentation — syntax, examples, and built-in linear algebra functions
• LaTeX Project — official LaTeX documentation
• Overleaf LaTeX Guide — quick tutorials for writing mathematical documents (easiest to work with!)
• Piazza — course discussion and homework questions

Grading Breakdown

• Assignments: 40%
• Two Midterm Exams: 30%
• Final Exam: 30%

Tips for Self-Guided Learning

• Use Ipsen for core theory, conditioning, and numerical stability
• Use Björck for deeper matrix-computation algorithms
• Use Horn and Johnson for matrix theory and rigorous reference material
• Practice MATLAB early; many concepts become clearer through computation
• Write solutions carefully in LaTeX, with clear notation and reproducible computations

This page summarizes the Summer 2026 offering. Course policies, schedules, and platform links may differ in future semesters.