Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia). Aptdo. 644, 48080 Bilbao, Spain
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.
This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator for two matrices and if and only if a vector
defined uniquely from the matrix is in the null space of a well-structured matrix defined as the Kronecker sum , 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 which extend the well-known sufficiency-type commuting result .