On k-potent matrices

Abstract

Abstract. The paper provides extensive and systematic investigations of k-potent complex matrices, with a particular attention paid to tripotent matrices. Several new properties of k-potent matrices are identified. Furthermore, some results known in the literature are reestablished with simpler proofs than in the original sources and often in a generalized form.

Inequalities and equalities for ℓ = 2 (Sylvester), ℓ = 3 (Frobenius), and ℓ > 3 matrices

“…Applying Theorem 6 with B = SS (2) M,N and recalling that S (2) M,N SS (2) M,N = S (2) M,N is valid, we get B 2 = B. Again, since S (2) M,N is a {2}-generalized inverse of S we have rank(SS (2) M,N C) = rank(S (2) M,N C). Analogously, rank(SS (2) M,N ) = rank(S (2) M,N ).…”

“…Similarly, the second one follows by induction on using (14) and Sylvester inequality. Some applications of formulae studied in this work were given in the recent paper [2]. In what follows, we present another application.…”

“…This unique matrix X is denoted by S (2) M,N . We also remind that, for certain special subspaces M and N , this generalized inverse S (2) M,N includes the classical inverses as particular cases: the Moore-Penrose inverse S † , the weighted Moore-Penrose inverse S † L,T (for L and T being hermitian positive definite matrices of appropriate sizes), the group inverse S # (whenever it exists), the Drazin inverse S D , and the weighted Drazin inverse S D W , among others. Proposition 14.…”

“…Let A ∈ C p×m , S ∈ C m×n of rank r, and C ∈ C m×q . Assume that M is a subspace of C n of dimension t ≤ r and N be a subspace of C m of dimension m − t such that S (2) M,N exists. If…”

“…Again, since S (2) M,N is a {2}-generalized inverse of S we have rank(SS (2) M,N C) = rank(S (2) M,N C). Analogously, rank(SS (2) M,N ) = rank(S (2) M,N ). Replacing these terms in (9) we arrive at equality (15).…”