Matrix Functions: Theory and Computation

This lecture explores the theory and computation of matrix functions, focusing on computing f(A) and particularly e^A. The instructor discusses the reasons for wanting f(A), different methods to define f(A) including diagonalization, polynomial interpolation, and contour integrals, and various approaches to compute f(A) such as the Schur-Parlett algorithm, polynomial and rational approximation, and discretized contour integrals. Special attention is given to the matrix exponential e^A, detailing ODE methods, scaling-and-squaring technique, contour integrals, and rational approximations. Common examples of matrix functions like the matrix exponential, sign function, square root, and power are also covered, along with applications in semidefinite programming and random walks.