Reachable Matrices by the Qr Iteration 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 QR algorithm with shift. A matrix B is said to be reachable by A if B = RQ + μI where A − μI = QR is the QR decomposition for some μ ∈ R. It is proved that in general the QR algorithm with shift is neither reflexive nor symmetric. It is further discovered that the reachable set from a given n × n matrix A forms 2 disjoint open loops if n is even and 2 disjoint components each of which is no longer a loop when n is odd.