Idempotency of linear combinations of an idempotent matrix and a tripotent matrix Original Research Article

The problem of characterizing situations, in which a linear combination C=c1A+c2B of an idempotent matrix A and a tripotent matrix B is an idempotent matrix, is thoroughly studied. In two particular cases of this problem, when either B or −B is an idempotent matrix, a complete solution follows from the main result in [Linear Algebra Appl. 321 (2000) 3]. In the present paper, a complete solution is established in all the remaining cases, when B is an essentially tripotent matrix in the sense that both idempotent matrices B1 and B2 constituting its unique decomposition B=B1−B2 are nonzero. The problem is considered also under the additional assumption that the differences A−B1 and A−B2 are Hermitian matrices. This obviously covers the case when A, B1, and B2 are Hermitian themselves, when the problem can be interpreted from a statistical point of view.