Two universal similarity factorization equalities for commutative involutory and idempotent matrices and their applications

Abstract

A square matrix A of order n is said to be involutory if A ² = I _n , and to be idempotent if A ² = A. In this article, we give two universal similarity factorization equalities for linear combinations of two commutative involutory and two idempotent matrices and their products. As applications, we derive some disjoint decompositions for these linear combinations, and use the disjoint decompositions to derive a variety of results on the determinants, ranks, traces, inverses, generalized inverses and similarity decompositions of these linear combinations. In particular, we present some collections of involutory, idempotent and tripotent matrices generated from these linear combinations.

Keywords:

• determinant
• disjoint idempotent decomposition
• idempotent matrix
• inverse
• involutory matrix
• linear combination
• rank
• tripotent matrix
• universal similarity factorization equality

AMS Subject Classifications:

• 15A23
• 15A27

Acknowledgements

The author would like to thank the referees for their careful reading and many constructive comments and suggestions on an earlier version of this article.