Some properties of commuting and anti-commuting m-involutions

We define an m-involution to be a matrix K ∈ ℂn×n for which Km = I. In this article, we investigate the class Sm (A) of m-involutions that commute with a diagonalizable matrix A ∈ ℂn×n. A number of basic properties of Sm (A) and its related subclass Sm (A, X) are given, where X is an eigenvector matrix of A. Among them, Sm (A) is shown to have a torsion group structure under matrix multiplication if A has distinct eigenvalues and has non-denumerable cardinality otherwise. The constructive definition of Sm (A, X) allows one to generate all m-involutions commuting with a matrix with distinct eigenvalues. Some related results are also given for the class (A) of m-involutions that anti-commute with a matrix A ∈ ℂn×n.