Perturbation analysis for the sign functions of regular matrix pairs

Abstract

This paper is devoted to the perturbation analysis for the matrix sign functions of the regular matrix pair. The first-order perturbation bounds and the global perturbation bounds are obtained. The results are illustrated by some numerical examples.If z ∈ C 0 , then sign(z) is undefined. Let the matrix Z ∈ C n×n and its Jordan decompositionwhere J + ∈ C r ×r and J − ∈ C (n−r )×(n−r ) have eigenvalues in C + and C − , respectively. The sign function of Z is defined to be [1]The simplest scheme for computing sign(Z ), in the case when Z is nonsingular, is the following Newton iteration applied to (sign(Z )) 2 = I :The iteration converges if Z has no eigenvalues on the imaginary axis and the asymptotic convergence rate is quadratic [1]. There are also several ways to define the matrix sign function sign(Z ) (see for instance [2,3]). There are two different ways to extend the matrix sign function for the matrix pairs. Let (A, B) be a regular matrix pair of order n and no eigenvalue of (A, B) lies on the imaginary axis. Then the pair (A, B) admits the following Jordan canonical form: there are n ×n nonsingular matrices X and Y such that (see [4])where J + ∈ C r ×r and J − ∈ C (n−r )×(n−r ) have eigenvalues in C + and C − , respectively. Then two matrix sign functions of (A, B) are defined as follows: