The generalized Schur complement in group inverses and (k ... - UPV

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The generalized Schur complement in group inverses and in (k + 1)-potent matrices 6c) A − ∈ A{1, 2}.d) A # exists and A − = A # .e) A − = A † .It is a well-known fact (see [5]) that S = D − CA − B is invariant under all choice ofA − ∈ A{1} if and only if the above conditions a) and b) hold. The following is the basicresult we need for deriving the remaining of the work. Observe that the next result allowsto express any power of the generalized Schur complement S by means of a very simpleformula which will permit to present further proofs in a simplified manner.Proposition 1 Let M be an idempotent matrix of the form (1) satisfying any of conditionsin (12) and let S ∈ C p×p be a generalized Schur complement as in (4). ThenS k+1 = S − kCE A F A B, for every positive integer k. (13)Proof: It is clear that M is a projector if and only ifA = A 2 + BC, B = AB + BD, C = CA + DC, D = CB + D 2 . (14)The proof is by induction on k. For k = 1, we proceed as in the proof of Theorem 2.1in [2] obtaining S 2 = S − CE A F A B. Suppose that the theorem is true for k thenS k+2 = S k+1 S = (S − kCE A F A B)S = S 2 − kCE A F A BS = S − CE A F A B − kCE A F A BS.It is sufficient to prove CE A F A BS = CE A F A B in order to finish the proof of the proposition.In fact, by (14),CE A F A BS = CE A F A B(D − CA − B)= CE A F A (BD − BCA − B)= CE A F A (B − AB − (A − A 2 )A − B)= CE A F A (I − A − AA − + A 2 A − )B= CE A F A (F A − AF A )B= CE A F A (I − A)F A B.Since F A (I − A)F A = F A , one has CE A F A BS = CE A F A B. The proof is then finished.We state without proof the following proposition which refers to the generalized Schurcomplement T defined in (5).
The generalized Schur complement in group inverses and in (k + 1)-potent matrices 7Proposition 2 Let M be an idempotent matrix of the form (1) satisfying any of thefollowing equivalent conditionsBD − DD − C = BD − C, BD − (I − DD − )C = O or B(I − D − D)D − C = Oand let T ∈ C m×m be a generalized Schur complement as in (5). ThenT k+1 = T − kB(I − D − D)(I − DD − )C, for every positive integer k.Under any of the conditions of the assumption (12) we have the following result relatedto spectral theory.Theorem 3 Let M be an idempotent matrix of the form (1) satisfying any of conditionsin (12) and let S ∈ C p×p be a generalized Schur complement as in (4). Then S(S−I) 2 = O.Proof: Denoting Q = CE A F A B, by Proposition 1 we haveS 2 = S − Q, S 3 = S − 2Q.Eliminating the matrix Q we get S 3 − 2S 2 + S = O. The proof is then finished. For each positive integer k and being S a generalized Schur complement of A in M asin (4), we define the following setP k = {S ∈ C p×p : S is defined as in (4) and S k+1 = S}.In next result, we characterize the elements of the set P k for each positive integer k.Theorem 4 Let M be an idempotent matrix of the form (1) satisfying any of conditionsin (12) and let S ∈ C p×p be a generalized Schur complement as in (4). The followingstatements are equivalent.1. S ∈ P 1 .2. There exists a positive integer k such that S ∈ P k .3. CE A F A B = O.4. S ∈ P k for all positive integer k.
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The generalized Schur complement in group inverses and (k ... - UPV

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