The generalized Schur complement in group inverses and (k + 1)-potent matrices

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1 The generalized Schur complement in group inverses and (k + 1)-potent matrices Julio Benítez Néstor Thome Abstract In this paper, two facts related to the generalized Schur complement are studied. The first one is to find necessary and sufficient conditions to characterize when the group inverse of a partitioned matrix can be expressed in the Schur form. The other one is to develop a formula for any power of the generalized Schur complement of an idempotent partitioned matrix and then to characterize when this generalized Schur complement is a (k + 1)-potent matrix. In addition, some spectral theory related to this complement is analyzed. Key words: Schur form; partitioned matrices; group inverse; projector. AMS subject classification: 15A09. 1 Introduction For a complex matrix M of the form [ A B M = C D ] C (m+p) (m+p), (1) it is well-known that if A C m m is nonsingular then the inverse of M is [ ] A M 1 = 1 + A 1 BS 1 CA 1 A 1 BS 1 S 1 CA 1 S 1 (2) This work was partially supported by Grant project BFM Departamento de Matemática Aplicada, Instituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia, 46071, Valencia, Spain. (jbenitez@mat.upv.es). Departamento de Matemática Aplicada, Instituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia, 46071, Valencia, Spain. (njthome@mat.upv.es). 1

2 The generalized Schur complement in group inverses and in (k + 1)-potent matrices 2 whenever there exists the inverse of the Schur complement S = D CA 1 B of A in M. The expression (2) is called the Schur form of the matrix M. Using the Schur complement T = A BD 1 C of D in M, being D C p p a nonsingular matrix, it can be stated the following similar expression to (2) for M 1 : [ ] M 1 T = 1 T 1 BD 1 D 1 CT 1 D 1 + D 1 CT 1 BD 1. Since the analysis is analogous, we only will refer to the formula (2) for M 1. The following conditions permit to define different generalized inverses for A C n m, a) AGA = A. b) GAG = G. c) GA = AG if m = n. d) (AG) = AG. e) (GA) = GA. The set of all matrices G C m n satisfying condition a) (or a) and b), resp.) is called the {1}-inverses (or {1, 2}-inverses, resp.) of A and it is denoted by A{1} (or A{1, 2}, resp.). The unique matrix G C n n satisfying a), b) and c) is called the group inverse of A and if it exists, it is denoted by A # (see [3]). The only matrix G C m n satisfying a), b), d) and e) is called the Moore-Penrose inverse of A and it is denoted by A. When A C n n and Range(A) = Range(A ), the matrix A is called an EP-matrix and in this case one has A = A # (see [3]). A square matrix A is called (k + 1)-potent if A k+1 = A. Throughout this paper, all appearing matrices must be conformable for the involved products. In this work, the expression [ ] A N = + A BS CA A BS S CA S (3) is considered and it is called the generalized Schur form for the matrix M given in (1) being S = D CA B, (4) for some fixed generalized inverses A A{1} and S S{1}, where S is called the generalized Schur complement of A in M. Analogously, we consider the generalized Schur complement T = A BD C (5)

3 The generalized Schur complement in group inverses and in (k + 1)-potent matrices 3 of D in M for any fixed generalized inverse D D{1}. Instead of the standard inverse M 1, Marsaglia and Styan (see [4]) extended the formula (2) for matrices in M{1}. Recently, Baksalary and Styan (see [1]) presented a characterization of the problem of when the generalized Schur form (3) coincides with the Moore-Penrose inverse of the matrix M. Now, we quote a result given in [1] which will be useful. Theorem 1 (Corollary 1 [1]) For a given matrix M of the form (1), the matrix N of the form (3) satisfies N M{1, 2} if and only if A A{1, 2}, S S{1, 2}, (I SS )C(I A A) = O, (I AA )B(I S S) = O and (I AA )BS C(I A A) = O. The last three conditions are independent of the choice of A A{1} and S S{1}. The paper is organized as follows. In Section 2 we study the group inverse of a partitioned matrix by means of the generalized Schur complement. Concretely, we characterize when the group inverse can be written in the generalized Schur form. Next, in Section 3, we characterize (k + 1)-potent generalized Schur complements constructed from a partitioned projector. We also derive a formula for any power of S which is easy to be used and some spectral theory of S is analyzed. 2 The group inverse using the generalized Schur complement In this section we present a result which characterizes when the group inverse of a partitioned matrix can be expressed in the generalized Schur form. As in [1], the following notation will be helpful: E A = I A A, F A = I AA, E S = I S S, F S = I SS, (6) where A A{1} and S S{1}. Theorem 2 For a given matrix M of the form (1), the matrix N of the form (3) satisfies N = M # if and only if A = A #, S = S #, F S C = O, F A B = O, BE S = O and CE A = O. Proof: By Theorem 1, we have that N M{1, 2} if and only if A A{1, 2}, S S{1, 2}, F A BE S = O, F S CE A = O and F A BS CE A = O. It is easy to see that [ ] [ ] AA MN = F A BS CA F A BS A F S CA SS, NM = A A BS CE A A BE S S CE A S. S

4 The generalized Schur complement in group inverses and in (k + 1)-potent matrices 4 The equality MN = NM is equivalent to the four conditions AA F A BS CA = A A A BS CE A (7) F A BS = A BE S (8) F S CA = S CE A (9) SS = S S (10) The conditions S S{1, 2} and (10) are equivalent to S = S #. Furthermore, from (10) we get E S = F S and so, from (7), (8) and (9) we get AA A A = A BS CE A + F A BS CA = A BF S CA + A BE S CA = A BE S CA + A BE S CA = O. Then the conditions A A{1, 2} and AA = A A are equivalent to A = A #. On the other hand, from (9) and using A A{1} and AA = A A we get F S CA A = S CE A A = S C(I A A)A = S C(A A A 2 ) = S C(A AA A) = O, and thus O = F S CA A = F S C(I E A ) = F S C F S CE A = F S C, (11) because F S CE A = O. From (8), (10) and S S{1} we get F A BS S = A BE S S = A B(I S S)S = A B(S SS S) = O, and then O = F A BS S = F A B(I E S ) = F A B F A BE S = F A B, because F A BE S = O. Replacing in (8) we get A BE S = O which implies AA BE S = O and by using (6), we get (I F A )BE S = O and since F A BE S = O then BE S = O.

5 The generalized Schur complement in group inverses and in (k + 1)-potent matrices 5 From (9) and (11) we get S CE A = O and then (6) implies that O = SS CE A = (I F S )CE A = CE A F S CE A = CE A, because F S CE A = O. The sufficiency follows by Theorem 1 and a simple computation. The proof is finished. It is clear that the above conditions derived here for the group inverse are very similar to those for the Moore-Penrose inverse given in [1]. It is not surprising because the group inverse coincides with the Moore-Penrose inverse for EP-matrices and then the following corollary can be established. Corollary 1 Under the notation in Theorem 2, if M is an EP-matrix then the following conditions are equivalent: a) N = M. b) N = M #. c) A = A, S = S, F S C = O, F A B = O, BE S = O and CE A = O. d) A = A #, S = S #, F S C = O, F A B = O, BE S = O and CE A = O. 3 (k + 1)-potent generalized Schur complement in projectors As in the previous section, for an idempotent matrix M partitioned as in (1), we shall denote A for a fixed generalized inverse matrix in A{1} and S = D CA B for the generalized Schur complement of A in M. In this section we derive the cases under which this matrix M has a generalized Schur complement to be (k + 1)-potent (i.e., S k+1 = S). Recalling the notation F A = I AA and E A = I A A, it is easy to see that CA AA B = CA B CA F A B = O CE A A B = O. (12) Any of the following conditions imply (see [3]) the three equivalent conditions in (12): a) Range(B) Range(A). b) Range(C ) Range(A ).

6 The generalized Schur complement in group inverses and in (k + 1)-potent matrices 6 c) 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 of A A{1} if and only if the above conditions a) and b) hold. The following is the basic result we need for deriving the remaining of the work. Observe that the next result allows to express any power of the generalized Schur complement S by means of a very simple formula 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 conditions in (12) and let S C p p be a generalized Schur complement as in (4). Then S 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 if A = 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 2.1 in [2] obtaining S 2 = S CE A F A B. Suppose that the theorem is true for k then S 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 Schur complement T defined in (5).

7 The generalized Schur complement in group inverses and in (k + 1)-potent matrices 7 Proposition 2 Let M be an idempotent matrix of the form (1) satisfying any of the following equivalent conditions BD DD C = BD C, BD (I DD )C = O or B(I D D)D C = O and let T C m m be a generalized Schur complement as in (5). Then T 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 related to spectral theory. Theorem 3 Let M be an idempotent matrix of the form (1) satisfying any of conditions in (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 have S 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 as in (4), we define the following set P 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 conditions in (12) and let S C p p be a generalized Schur complement as in (4). The following statements are equivalent. 1. S P 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.

8 The generalized Schur complement in group inverses and in (k + 1)-potent matrices 8 Proof: The implications 1 2 and 4 1 are trivial. The implications 2 3 and 3 4 follow directly from Proposition 1. The proof is finished. Note that the same condition characterizing the elements of P 1 is valid to characterize those of P k for each positive integer k. In other words, P 1 = P k for each positive integer k, which a priori is not evident. From Theorem 3, it is clear that the spectrum of S satisfies σ(s) {0, 1}. Moreover, some extra information about the generalized Schur complement S is obtained in the following result. Theorem 5 Let M be an idempotent matrix of the form (1) satisfying any of conditions in (12) and let S C p p be a generalized Schur complement as in (4). The following statements hold. 1. If zero is not an eigenvalue of S then (S I) 2 = O. Moreover, A 2 = A, BC = O, BD = B, AB = O, DC = C, CA = O and D(D I) 2 = O. 2. If one is not an eigenvalue of S then S = O. Proof: By Theorem 3 one has S(S I) 2 = O. If 0 is not an eigenvalue of S then S is nonsingular and thus (S I) 2 = O. Expanding this last equation and using S 2 = S CE A F A B we obtain S + CE A F A B = I, i.e., Premultiplying by B one has D CA B + CE A F A B = I. (15) BD BCA B + BCE A F A B = B (16) and using (14) we get BC(E A F A A )B = AB. From (14) and postmultiplying this last equality by C we get (A A 2 )A BC = ABC because AE A = O. Since BC = A A 2 then (A A 2 )A (A A 2 ) = A(A A 2 ). Expanding the left side of this equality we get (A A 2 )A (A A 2 ) = A(I A)A (I A)A = A(A A A AA + A)A = A 2A 2 + A 3, since AA A = A and thus A 2 = A. Using (14) we get BC = O and from (16) we obtain BD = B. Again from (14) we get AB = O. Postmultiplying (15) by C we obtain DC = C and by (14), CA = O. Finally, since D D 2 = CB = CBD = (D D 2 )D = D 2 D 3

9 The generalized Schur complement in group inverses and in (k + 1)-potent matrices 9 then D(D I) 2 = O. The statement 2. follows directly from Theorem 3. The affirmation on D in the condition 1. of the previous theorem can not be improved. The example [ A B M = C D ] [ 1 0, A = 0 0 ] [ 0 0, B = b a ] [ 0 a, C = 0 b ] [ 1 + ab a 2, D = b 2 1 ab with a 2 + b 2 = 1, satisfies all the statements in the condition 1 of Theorem 5 and, however, D 2 D. Moreover, for generalized Schur complements S satisfying S k+1 = S for certain k, we derive the following result. Theorem 6 Let M be an idempotent matrix of the form (1) satisfying any of conditions in (12) and let S C p p be a generalized Schur complement as in (4). If there exists a positive integer k such that S k+1 = S then the following statements are equivalent: 1. Zero is not eigenvalue of S (i.e., S is nonsingular). 2. S = I. 3. D = I and CA B = O. Proof: 1 2: By Proposition 1 we get S k+1 = S kce A F A B, S k = S (k 1)CE A F A B. (17) Since S k+1 = S, we obtain CE A F A B = O. Since S k+1 = S and S is nonsingular then S k = I and from (17), S = I. The implications 2 3 and 3 1 follows as in Theorem 3.1 of [2]. Proposition 2 permits to give analogous results corresponding to the Theorems 3, 4, 5 and 6 for the generalized Schur complement (5) denoted by T. Again, we remark that the simplicity of this last proof is due to the simple formula (13). In view that Theorem 6 extends Theorem 3.1 in [2] using the same condition 3, a similar argument to those in Theorem 3.2 permits to compute the eigenvectors corresponding to the zero eigenvalue of the generalized Schur complements S and T. We derive the result without proof. Theorem 7 Let M C (m+p) (m+p) be an idempotent matrix partitioned as in (1) and let S C m m be the generalized Schur complement defined in (4). Denote x C (m+p) 1 by [u : v ]. If Range(B) Range(A) and Range(C ) Range(A ) ].

10 The generalized Schur complement in group inverses and in (k + 1)-potent matrices 10 and then A I or BD C O 1. Mx = 0 and u 0 implies Su = Su = 0, u 0 and v = D Cu implies Mx = 0. References [1] J.K. Baksalary and G.P.H. Styan. Generalized inverses of partitioned matrices in Banachiewicz-Schur form, Linear Algebra Appl. 354 (2002) [2] J.K. Baksalary, O.M. Baksalary and T. Szulc. Properties of Schur complements in partitioned idempotent matrices, Linear Algebra Appl. 379 (2004) [3] A. Ben-Israel and T. Greville. Generalized inverses: theory and applications, Wiley, New York, (1974). [4] G. Marsaglia and G.P.H. Styan. Rank Conditions for generalized inverses of partitioned matrices, Sankhyā Ser. A (1974) [5] C.R. Rao and S.K. Mitra. Generalized inverses of matrices and its applications, Wiley, New York, (1971).