Please use this identifier to cite or link to this item: https://doi.org/10.1137/040617844
Title: Reachable matrices by a QR step with shift
Authors: Chu, D. 
Chu, M.
Keywords: Analytic QR decomposition
QR algorithm
Reachable matrices
Reflexivity
Shift
Symmetry
Transitivity
Issue Date: 2006
Citation: Chu, D., Chu, M. (2006). Reachable matrices by a QR step with shift. SIAM Journal on Applied Dynamical Systems 5 (1) : 91-107. ScholarBank@NUS Repository. https://doi.org/10.1137/040617844
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 + μI, where A - μI = QR is the QR. decomposition for some μ ε 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 × 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. © 2006 Society for Industrial and Applied Mathematics.
Source Title: SIAM Journal on Applied Dynamical Systems
URI: http://scholarbank.nus.edu.sg/handle/10635/104030
ISSN: 15360040
DOI: 10.1137/040617844
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