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 × 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.