(AA-A = A) and A + be the Moore-Penrose inverse of A [4]. A matrix A is called EP r

Transcription

1 HOUSTON JOURNAL OF MATHEMATICS, Volume 9, No. 1, ON SUMS OF EP MATRICES AR. Meenakshi* let A* 0. Introduction. Throughout we shall deal with n X n complex matrices. For A, denote the conjugate transpose of A; let A- be a generalized inverse (AA-A = A) and A + be the Moore-Penrose inverse of A [4]. A matrix A is called EP r if rk(a) = r and N(A) = N(A*) or R(A) = R(A*) where rk(a) denotes the rank of A; N(A) and R(A) denote the null space and range space of A respectively. For further properties of EP matrices one may refer to [1,3,5]. In this paper we study the question of when a sum of EP matrices is EP. In Section 1, we give necessary and sufficient conditions for sum of EP matrices to be EP. In Section 2, it is shown that sum and parallel sum of parallel summable [4, page 188] EP matrices are EP. 1. Sums of EP matrices. THEOREM 1. Let A i (i = 1 to m) be EP matrices. Then A = 2A i is EP if any one of the follo wing equivalent conditions hold.' (i) N(A) _C N(Ai)for each i, A 1 (ii) rk A 2 = rk(a). PROOF. (i) * (ii) N(A) C_ N(Ai) for each i implies N(A) C_ ChN(Ai). Since N(A) = N(ZAi)_D N(Ai)r3 N(A2)r3---C3 N(Am) it follows that N(A)D C3 N(Ai). Hence N(A) = r3 N(Ai) = N A2 m *Supported in part by NSERC of Canada Grant No. A

2 64 AR. MEENAKSHI Therefore rk(a) = rk A 1 ' A 2 ß and (ii) holds. Conversely, since N A 1 A 1 A 2 = Ch N(Ai) C_ N(A); rk A2 = rk(a) implies N(A) = r3 N(Ai). Hence, N(A) c_ N(Ai) for each i and (i) holds. Sinceach A i is EP, N(A i) = N(A *) for each i. N(A) C_ N(Ai) for each i =} N(A) _C r3 N(Ai) = r3 N(A' ) C- N(A*) and rk(a) = rk(a*). Hence N(A) = N(A*). Thus A is EP. Hence the theorem. REMARK 1. In particular if A is nonsingular the conditions automatically hold and A is EP. Theorem 1 fails if we relax the condition on the Ai's. EXAMPLE1. A= [ 0] is EP; B= [l 08] is not EP. A+B: [l 8 ] is not EP. However, N(A + B) C- N(A) and N(A + B) C- N(B)' rk[ ] = rk(a + B). REMARK 2. If rank is additive, that is rk(a) --!;rk(ai), then by Theorem 11 in [2], R(A i) (q R(Aj) = {0}, i :/=j, which implies N(A) C_ N(Ai) for each i, hence A is EP. The conditions given in Theorem 1 are weaker than the condition of rank additivity can be seen by the following example. EXAMPLE 2. Let A=[ ] and B=[ ]. A, Band(A+B) are EP matrices. Conditions (i) and (ii) in Theorem 1 hold but rk(a + B) :/= rk(a) + rk(b). THEOREM 2. Let A i (i = 1 to m) be EP matrices such that then A =!;A i is EP. Ei:#jA]Aj = 0, PROOF. Since Eig=jAtAj = 0, A*A = (EA )(EA i) = EA;A i N(A) = N(A*A) = N(EA A i)

3 ON SUMS OF EP MATRICES 65 = N. A2 A 1 A 2 = N = N(A1) C3 N(A2) C3---C3 N(Am). Hence N(A) _C N(Ai) for each i. Sinceach A i is EP, A is EP by Theorem 1. instance, REMARK 3. Theorem 2 fails if we relax the condition that Ai's are EP. For A = -1 and B = 0 are not EP; A + B is also not EP. However, A*B + B*A = 0. REMARK 4. The condition given in Theorem 2 implies those in Theorem 1, but not conversely. This can be seen by the following: EXAMPLE 3. Let A= [I 1 ] and B= [_01 ]. A and B are EP matrices. N(A + B) _C N(A) and N(B). But, A*B + B*A :P- 0. REMARK 5. That the conditions given in Theorem 1 and Theorem 2 are only sufficient forthe sum of EP matrices to be EP but not necessary is illustrated by the following: EXAMPLE 4. Let A=[ ] and B=[_ ]. A and B are EP 2. Neither the conditions in Theorem 1 nor in Theorem 2 hold. However A + B is EP. If A and B are EP matrices, by Theorem 1 in [3], A* = K1A and B* = K2B where K 1 and K 2 are nonsingular n X n matrices. If K 1 = K 2, then (A + B) is EP [ 1, page 76]. If (K 1 - K2) is nonsingular then the above conditions are also necessary for the sum of EP matrices to be EP is given in the following theorem. THEOREM 3. Let A* = K1A and B* = K2B such that (K 1 - K2) is a nonsingular matrix. Then (A + B) is EP N(A + B) _C N(B). PROOF. Since A*=K1A and B*=K2B by Theorem 1 in [3] Aand B are EP matrices. Since N(A + B) _C N(B) we can see that, N(A + B) _C N(A). Hence by

4 66 AR. MEENAKSHI Theorem 1, (A + B) is EP. Conversely, let us assume that (A+ B) is EP. Now by Theorem I in [3], A*+B*= (A+B)* = G(A+B) œor some nxn matrix G. Hence, KiA+K2B = G(A + B). This implies KA = HB, where K = K 1 - G and H = G - K2, (K + H)A = H(A + B) and (K + H)B = K(A + B). By hypothesis, K + H = K 1 - K 2 is nonsingular. N(A + B) _C N(H(A + B)) = N((K + H)A) = N(A). N(A + B) _C N(K(A + B)) = N((K + H)B) = N(B). Thus (A + B) is EP N(A + B) _C N(A) and N(B). Hence the theorem. REMARK 6. The condition (K 1 - K2) to be nonsingular is essential in Theorem 3 is illustrated in the œollowing: EXAMPLES. A= [0 ] and B= [ ] are both symmetric hence EP; 00 K 1 = K 2. A + B = [0 1 ] is EP but N(A + B) N(A) or N(B). Thus Theorem 3 œails. 2. Parallel summable EP matrices. In this section we shall show that sum and parallel sum oœ parallel summable EP matrices are EP. First we shall give the definition and some properties oœ parallel summable matrices as in [4, page 188]. DEFINITION1. A and B are said to be parallel summ. able (p.s.) it N(A + B) C N(B) and N(A + B)* C N(B*) or equivalently N(A + B) C N(A) and N(A + B)* _C N(A*). DEFINITION 2. Iœ A and B are parallel summable then parallel sum oœ A and B denoted by A B is defined as A B = A(A + B)-B. (The product A(A + B)-B is invariant for all choices oœ generalized inverse (A + B)- oœ (A + B) under the conditions that A and B are parallel summable [4, page 21].) PROPERTIES. Let A and B be a pair oœ parallel summable (p.s.) matrices. Then the œollowing hold: p. 1 A -i- B = B -i- A p.2 A* and B* are p.s. and (A -_+ B)* = (A* B*) p.3 If U is nonsingular then UA and UB are p.s. and UA - UB = U(A B). p.4 R(A B) = R(A) N R(B); N(A +_- B) = N(A) + N(B)

5 ON SUMS OF EP MATRICES 67 defined. p.5 (A +_- B) -T- E = A +_- (B +_- E) if all the parallel sum operations involved are LEMMA 1. Let A and B be EP matrices. Then A and B are p.s. * N(A + B) _C N(A). PROOF. A and B are p.s. = N(A+ B)_C N(A) follows from Definition 2. Conversely, if N(A + B) _C N(A) then N(A + B) _C N(B). Since A and B are EP matrices, by Theorem 1, (A+B) is EP. Hence, N(A+B)* = N(A+B) = N(A) C N(B)= N(A*) C N(B*). Therefore, N(A + B)* C- N(A*) and N(A + B)* C N(B*). By hypothesis N(A + B) _C N(A). Hence A and B are p.s. REMARK 7. Lemma 1 fails if we relax the condition that A and B are EP A=[0 0 ] isep. B=[1 0 ] is notep. N(A + B) _c N(A) and N(B) but N(A + B)* N(A*) or N(B*). Hence A and B are not parallel summahie. EP. THEOREM 4. Let A and B be p.s. EP matrices. Then, (A _$ B) and (A + B) are PROOF. Since A and B are p.s. EP matrices by Lemma 1, N(A + B) C_ N(A) and N(A + B) _C N(B). Now, the fact that (A + B) is EP follows from Theorem 1. R(A _ B)* = R(A* - B*) (By p. 2) = R(A*) rh R(B*) (By p.4) = R(A) rh R(B) (A and B are EP) = R(A _$ B) (By p.4). Thus (A -_+ B) is EP whenever A and B are EP. Hence the theorem. REMARK 8. The sum and parallel sum of p.s. EP matrices is EP. COROLLARY 1. Let A and B be EP matrices such that N(A + B) C N(B). If C is EP commuting with both A and B, then C(A + B) and C(A - B) = CA_½ CB are EP. PROOF. By Theorem 1, (A + B) is EP. Since C commutes with A, B and (A + B), by Theorem (1.3) in [1], CA, CB and C(A + B) are EP. Now by Theorem 4, CAff- CB - is EP. By Theorem in [4] C(A - B) = CA - CB. Since CA - CB is EP, C(A 4- B) - is EP. Hence the result.

6 68 AR. MEENAKSHI DEFINITION 3. Let M = [ )] be a n X n matrix. The Schur complement of A in M, denoted by M/A is defined as D - CA'B, where A- is a generalized inverse of A [ 2, page 291 ]. THEOREM 5. Let A be EPrl and B be EPr2 matrices such that N(A + B) _C_ N(B). Then there exists a 2n X 2n EP r matrix M such that the $chur co nplement of C in M is EP, where r = r 1 + r 2 and C = A + B. PROOF. Since A and B are EP, by Theorem 1 in [3] there exists unitary matrices U and V of order n such that A = U*DU and B = V*EV, where D = [O H ], H is r 1 X r I nonsingular E= [O K ( ].K is r 2 X r 2 nonsingular. Let us define P = [IVJ 0 I ], P is nonsingular. E V* U* E V 0 ia+b U*D P*[0 D 0]P=[0 I ][0 I ][U I ] =LDU D ] C AU* = [UA UAU *] = M. Mis2n X2nmatrixandrkM=rkE+rkD=r 1 +r 2=r. I n 0 Let us define Q = [UA+A In ], Q is nonsingular. Since A is EP, AA + = A+A, and UAU* is EP. We can write M as, M = Q*[g UAU* 0 ] Q' Since B and UAU* are EP, Q is nonsingular, M is EP. Since M is of rank r, M is EP r- Thus we have proved the existence of the EP matrix M. Now C = A + B is EP follows from Theorem 1. Since N(C) C N(A) = N(UA) and N(C*) _C N(A*) = N(AU*)*, by the lemma in [4, page 21 ], A = AC-C = CC-A and (UA)C-(AU*) is invariant for all choices of C-. The Schur Complement of C in M is M/C = UAU* - UAC-AU* = UAU* - U(A + B)C'AU* + UBC-AU* = UAU* - UCC-AU* + UBC-AU* = UAU* - UAU* + UB(A + B)-AU* = U(A 4 B)U*. -

7 ON SLIMS OF EP MATRICES 69 Since A and B are EP, by Theorem 4, A -[ B is EP, therefore M/C = U(A B)U* is also EP. Hence the theorem. REMARK 9. In a special case if A and B are EP matricesuch that A + B = I n, then AB = A -7- B = B/- A = BA is EP. However this fails if we relax the conditions on A and B. For instance, A=[1 1] isep 2andB=[ Here, AB = BA is not EP, however A + B = 12. _01 ] is notep. 3. Application. We can use Theorem 1 to see whether a product of two rectangular matrices is EP. Let L = [A:B] and M = [i ] where A, B, C, D are n X n EP matrices, such that AC = CA and BD = DB. If N(LM) C_ N(AC) then LM is EP. Since A and C are commuting EP matrices by Theorem (1.3) in [ 1 ] AC is EP. Similarly BD is EP. N(AC + BD) = N(LM) C N(AC). Hence by Theorem 1, LM = AC + BD is EP. For example, suppose L = [ 00 ] = [A:B] and M= - j [D 1' LM = AC + BD = A + D. L and M are of rank 2, N(LM) = N(A). Hence LM is EP 1. ACKNOWLEDGEMENT. The author wishes to thank D. Z. Djokovic and P1. Kannappan for many helpful discussions. REFERENCES 1. I.J. Katz and M. H. Pearl, On EP r and normal EP r matrices, J. Res. NBS, 70(1966), G. Marsaglia and G. P. H. Styan, Equalities and inequalities for ranks of matrices, Linear and Multilinear Alg., 2(1974), M. H. Pearl, On normalandep r matrices, Michigan Math. J., 6(1959), C. R. Rao and S. K. Mitra, Generalized inverse of matrices and its applications, Wiley & Sons, New York, H. Schwerdtfeger, Introduction to linear algebra and the theory of matrices, P. Noordhoff, Groningen, , Mariappanagar Tamilnadu, India Received September 15, 1981

8