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 4The equality MN = NM is equivalent to the four conditionsAA − − 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 getAA − − 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 getF 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 thusO = 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 getF A BS − S = A − BE S S= A − B(I − S − S)S= A − B(S − SS − S)= O,and thenO = 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 = Oand by using (6), we get (I − F A )BE S = O and since F A BE S = O then BE S = O.
The generalized Schur complement in group inverses and in (k + 1)-potent matrices 5From (9) and (11) we get S − CE A = O and then (6) implies thatO = 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 similarto those for the Moore-Penrose inverse given in [1]. It is not surprising because the groupinverse coincides with the Moore-Penrose inverse for EP-matrices and then the followingcorollary can be established.Corollary 1 Under the notation in Theorem 2, if M is an EP-matrix then the followingconditions 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 projectorsAs in the previous section, for an idempotent matrix M partitioned as in (1), we shalldenote A − for a fixed generalized inverse matrix in A{1} and S = D − CA − B for thegeneralized Schur complement of A in M. In this section we derive the cases under whichthis 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 thatCA − 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 ∗ ).
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