Abstract
Centrosymmetric matrices are matrices that commute with the permutation matrix J, the matrix with ones on its cross-diagonal. This paper generalizes the concept of centrosymmetry, and considers the properties of matrices that commute with an arbitrary permutation matrix P, the P-commutative matrices. In particular,it focuses on two related classes of matrices: inflation matrices and $ZME$-matrices. The structure of P-commutative inflators is determined, and then this is used to characterize the P-commutative $ZME$-matrices. Centrosymmetric matrices in these classes are presented as a special case.
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