Powers of matrices and idempotency

Abstract

In this paper, we establish the following results: Let A be a square matrix of rank r. Then (a) (A+A ∗) 2 is idempotent of rank r, and tr r A (defined as the sum of the principal minors of order r in A) is one iff A is Hermitian idempotent. (b) A s = A t for some positive integers s≠ t, and tr A= rank A iff A is idempotent. (c) A(A∗A) s= A(AA ∗) t for some integers s≠ t iff AA ∗=A ∗A is idempotent, while A(A ∗A) s= A(AA ∗) s for some integers s≠0 iff AA ∗=A ∗A . (d) A(A ∗A) s=A ∗ (AA ∗) t for some integers s≠ t and rank A= tr A iff A is Hermitian idempotent, while A(A ∗A) s= A ∗(AA ∗) s for some integer s iff A is Hermitian. Here A ∗ indicates the conjugate transpose of A, and P - α is defined iff ( P +) α =( P α ) + for all positive integers α and P + is the Moore-Penrose inverse of P.