The Chebyshev Polynomials of a Matrix

Abstract

A Chebyshev polynomial of a square matrix A is a monic polynomial of specified degree that minimizes ||p(A)||(sub2). The study of such polynomials is motivated by the analysis of Krylov subspace iterations in numerical linear algebra. An algorithm is presented for computing these polynomials based on reduction to a semidefinite program which is then solved by a primal-dual interior point method. Examples of Chebyshev polynomials of matrices are presented, and it is noted that if A is far from normal, the lemniscates of these polynomials tend to approximate pseudospectra of A.