SOME PROPERTIES OF COMMUTING AND ANTI-COMMUTING m-INVOLUTIONS

We define an m-involution to be a matrix K is an element of C-n x n for which K-m = I. In this article, we investigate the class S-m (A) of m-involutions that commute with a diagonalizable matrix A is an element of C-n x n. A number of basic properties of S-m (A) and its related subclass S-m (A, X) are given, where X is an eigenvector matrix of A. Among them, S-m (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 S-m (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 (S) over tilde (m) (A) of m-involutions that anti-commute with a matrix A is an element of C-n x n.