Idempotency of linear combinations of three idempotent matrices, two of which are disjoint

Abstract

Given nonzero idempotent matrices A 1, A 2, A 3 such that A 2 and A 3 are disjoint, i.e., A 2 A 3= 0 = A 3 A 2 , the problem of characterizing all situations, in which a linear combination C =c 1 A 1+c 2 A 2+c 3 A 3 is an idempotent matrix, is studied. The results obtained cover those established by J.K. Baksalary, O.M. Baksalary, and G.P.H. Styan (Linear Algebra Appl. 354 (2002) 21) under the additional assumption that c 3=− c 2, i.e., in the particular case where C =c 1 A 1+c 2( A 2− A 3) is actually a linear combination of an idempotent matrix A 1 and a tripotent matrix A 2− A 3 .