Chapter 9: Functions of Matrices

Abstract:

Let p be a polynomial

then the corresponding matrix function of an n × n matrix A over ℂ can be defined by with the convention A⁰ = I_n. If a function f of a complex variable z has a MacLaurin series expansion which converges for |z| < R (R > 0), then the matrix series converges, provided A is square and each of its eigenvalues has absolute value less than R. The most used matrix function is exp(A) defined by