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Zbl 1183.15014
de la Sen, M.
On the necessary and sufficient condition for a set of matrices to commute and some further linked results. (English)
[J] Math. Probl. Eng. 2009, Article ID 650970, 24 p. (2009). ISSN 1024-123X; ISSN 1563-5147/e

Summary: The author investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator $[A,B]$ for two matrices $A$ and $B$ equals zero 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 $A\oplus ( - A^{\ast })$, 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 $[A,f(A)]=0$.

MSC 2000:: *15A27 Commutativity of matrices
15A16

Keywords: commutativity; commutator; Kronecker sum; functions of matrices

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