A Note on k-Commutative Matrices | Zendy

Donald W. Robinson | Zendy

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A Note on k-Commutative Matrices

Author(s) -

Donald W. Robinson

Publication year - 1961

Publication title -

journal of mathematical physics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.708

H-Index - 119

eISSN - 1089-7658

pISSN - 0022-2488

DOI - 10.1063/1.1724222

Subject(s) - mathematics , commutative property , idempotence , integer (computer science) , matrix (chemical analysis) , field (mathematics) , pure mathematics , square matrix , combinatorics , polynomial , commutative ring , square (algebra) , finite field , discrete mathematics , algebra over a field , symmetric matrix , eigenvalues and eigenvectors , mathematical analysis , geometry , physics , materials science , quantum mechanics , computer science , composite material , programming language

Let A and B be square matrices over a field in which the minimum polynomial of A is completely reducible. It is shown that A is k commutative with respect to B for some non‐negative integer k if and only if B commutes with every principal idempotent of A. The proof is brief, simplifying much of the previous study of k‐commutative matrices. The result is also used to generalize some well‐known theorems on finite matrix commutators that involve a complex matrix and its transposed complex conjugate.

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