Reachable Matrices by a QR Step with Shift

Abstract

One of the most interesting dynamical systems used in numerical analysis is the $QR$ algorithm. An added maneuver to improve the convergence behavior is the $QR$ iteration with shift which is of fundamental importance in eigenvalue computation. This paper is a theoretical study of the set of all isospectral matrices "reachable" by the dynamics of the $QR$ algorithm with shift. A matrix B is said to be reachable by A if $B = RQ + \mu I$, where $A - \mu I = QR$ is the $QR$ decomposition for some $\mu \in \mathbb{R}$. It is proved that in general the $QR$ algorithm with shift is neither reflexive nor symmetric. Examples are given to demonstrate that this relation is neither transitive nor antisymmetric. It is further discovered that the reachable set from a given $n \times n$ matrix A forms $2^{n-1}$ disjoint open loops if n is even and $2^{n-2}$ disjoint components each of which is no longer a loop when n is odd.