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

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.