Idempotency of linear combinations of two idempotent matrices

Notes on linear combinations of two tripotent , idempotent , and involutive matrices that commute

The aim of this paper is to provide alternate proofs of all the results of our previous paper [2] in the particular case when the given two matrices A1 and A2 in the linear combination A = c1A1 + c2A2 commute.

The Group Involutory Matrix of the Combinations of Two Idempotent Matrices

On Nonsingularity of Linear Combinations of Tripotent Matrices

Let T1 and T2 be two commuting n × n tripotent matrices and c1, c2 two nonzero complex numbers. The problem of when a linear combination of the form T = c1T1 + c2T2 is nonsingular is considered. Some other nonsingularitytype relationships for tripotent matrices are also established. Moreover, a statistical interpretation of the results is pointed out.

Ela on the Group Inverse of Linear Combinations of Two Group Invertible Matrices

hold. If such matrix X exists, then it is unique, denoted by A, and called the group inverse of A. It is well known that the group inverse of a square matrix A exists if and only if rank(A) = rank(A) (see, for example, [1, Section 4.4] for details). Clearly, not every matrix is group invertible. It is straightforward to prove that A is group invertible if and only if A is group invertible, and ...

On the group inverse of linear combinations of two group invertible matrices