Spring 2025 STAT 30960

Course: STAT 30960=CAAM 30960

Title: Matrix Calculus and Matrix Analysis


Instructor: Lek-Heng Lim

Class Schedule: Sec 1: MW 3:00 PM–4:20 PM in TBA

Description: Just as basic real analysis concerns real-valued functions of one real variable and basic complex analysis concerns complex-valued functions of one complex variable, this class is about matrix-valued functions of one matrix variable. We will revisit the same basic concepts in real and complex analysis but now in the context of matrix analysis: continuity (standard, uniform, Lipchitz, Holder, etc), differentiability, higher-order differentiability, analyticity, convexity, monotonicity, sequences, series (particularly Taylor, Fourier, Dirichlet), integration (Bochner, Pettis, Cauchy, etc), and more. We will also develop a corresponding matrix calculus for differentiating matrix-valued functions of matrix variables.

We will study several important special cases: (1) functions of square matrices defined by power series: matrix exponential, matrix logarithm, matrix square root, etc; (2) real-valued matrix functions: trace inverse, log determinant, determinant, etc; (3) functions defined by determinants: minors, cofactors, adjugates, compounds, etc; (4) functions defined by sums, products, and elementary symmetric polynomials of eigenvalues and singular values.

We will discuss applications to: (a) maximum likelihood estimation; (b) semidefinite programming; (c) optimal experimental designs; (d) optimization of matrix-valued functions; (e) computations of matrix functions; (f) conditioning of matrix decompositions. Attention will be on with matrix functions originating from engineering and scientific applications. Everything that we do will ultimately be motivated by real-world problems.