On linear combinations of two tripotent, idempotent, and involutive matrices

Abstract

Let A = c(1)A(1) + c(2)A(2), where c(1), c(2) are nonzero complex numbers and (A(1), A(2)) is a pair of two n x n nonzero matrices. We consider the problem of characterizing all situations where a linear combination of the form A = c(1)A(1) + c(2)A(2) is (i) a tripotent or an involutive matrix when A(1) and A(2) are commuting involutive or commuting tripotent matrices, respectively, (ii) an idempotent matrix when A(1) and A(2) are involutive matrices, and (iii) an involutive matrix when A(1) and A(2) are involutive or idempotent matrices. (C) 2007 Elsevier Inc. All rights reserved.