Math 104: Applied Matrix Theory

Math 104
Applied Matrix Theory
Fall 2025

Instructor
Emmanuel Candes
144 Sequoia Hall

Lectures
Tuesday, Thursday
10:30-11:50 a.m.
Building 370, Rm 370

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Description: The aim of this course is to introduce the key mathematical ideas in matrix theory, which are used in modern methods of data analysis, scientific computing, optimization, and merely all quantitative fields of science and engineering. While the choice of topics is motivated by their use in various disciplines, the course will emphasize the theoretical and conceptual underpinnings of this subject, just as in other (applied) mathematics course.

Prerequisite: Math 51, CS 106, and either Math 52 or Math 53. We expect all students to be familiar with the following notions:

vector operations: dot product, cross product

matrix operations: matrix-matrix and matrix-vector multiplications

partial derivatives and the chain rule of vector calculus

definition of eigenvalue and eigenvector

3-by-3 determinants

Syllabus:

Matrices, vectors and their products (review)

Matrices as linear transformations

Rank of a matrix, linear independence and the four fundamental subspaces of a matrix

Orthogonality and isometries

The QR decomposition

Eigenvalues and the spectral decomposition of symmetric matrices

The singular value decomposition and its applications

The conditioning of a matrix

Least squares problems

Algorithms for solving systems of linear equations and least-squares problems

Iterative methods for solving linear systems: the method of conjugate gradients

Applications (mostly to data science): e.g. multivariate linear regression and principal component analysis