Some results on idempotency and tripotency of linear combinations of matrices

It is important to consider the problem of linear combinations of idempotent and tripotent matrices in matrix theory and statistics.Let A and B be n×n nonzero complex matrices.Denote a linear combination of the two matrices by P=c1A+c2B,where c1 and c2 are nonzero complex numbers.In this paper,if AB=BA,then we give(i) the sufficient and necessary conditions for idempotency of the matrix P,where A is an idempotent matrix and B is an arbitrary matrix;(ii) the sufficient and necessary conditions for tripotency of the matrix P,where A is a tripotent matrix and B is an arbitrary matrix.Moreover,based on the above results,we have(i) the sufficient and necessary conditions for idempotency of the matrix P,where A is an idempotent matrix and B is a tripotent matrix or an idempotent matrix which is commutative with A;(ii) the sufficient and necessary conditions for tripotency of the matrix P,where A and B are tripotent matrices that are commutative with each other.