Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 650970, 24 pages
doi:10.1155/2009/650970
Research Article

On the Necessary and Sufficient Condition for a Set of Matrices to Commute and Some Further Linked Results

M. De La Sen

Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia). Aptdo. 644, 48080 Bilbao, Spain

Received 5 January 2009; Revised 4 May 2009; Accepted 9 June 2009

Academic Editor: Angelo Luongo

Copyright © 2009 M. De La Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator [A,B]=0 for two matrices A and B if and only if a vector v(B) defined uniquely from the matrix B is in the null space of a well-structured matrix defined as the Kronecker sum A⊕(−A∗), which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning the commutators of certain matrices with functions of matrices f(A) which extend the well-known sufficiency-type commuting result [A,f(A)]=0.