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Displacement structure of W-weighted Drazin inverse Ad,W and its perturbation
Applied Mathematics and Computation 162 (2005) 403–419 www.elsevier.com/locate/amc
Displacement structure of W -weighted Drazin inverse Ad;W and its perturbation Mei Qin *, Guanglin Wang Institute of Mathematics, Fudan University, Shanghai 200433, PR China
Abstract It is well known that matrices with a UV -displacement structure possess generalized inverse with a VU-displacement structure. Estimations for the displacement rank of Ad;W U VAd;W are presented, where Ad;W is the W -weighted Drazin inverse of A. We extend the results for the Drazin inverse and group inverse, respectively. In the end, the perturbation of W -weighted Drazin inverse Ad;W is studied. 2004 Elsevier Inc. All rights reserved. Keywords: Displacement; W -weighted Drazin inverse; Perturbation
1. Introduction and preliminaries The present paper is to study the generalized inverse of a structured matrix. To begin with we recall some facts concerning the regular inversion of structured matrices, which are motivated by [6,7]. If the rank of a matrix’s displacement is small, fast algorithms for the matrix are available. A matrix is called matrices with displacement structure [4,5,9] if and only if the rank of the matrix AU VA or A VAU is small compared with the order of the matrix A. The rank of AU VA is said to be Sylvester UV -displacement rank and the rank of A VAU is called the Stein UV -displacement rank of A, since A is the solution of a Sylvester or Stein equations, respectively.
*
Corresponding author. E-mail address: [email protected] (M. Qin).
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.140
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M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
As we all know that fast inversion algorithms for matrix A can be constructed if A is matrix with displacement structure. We are interested in the generalized inverse with as small as possible a displacement rank. We present the estimate for the rank of Ad;W U VAd;W , where Ad;W is the W -weighted Drazin inverse of A. Then we give an explicit estimation for general displacement. Our results cover the previous result [3,4,20] for the Drazin inverse and group inverse, respectively. If A and W are any complex matrices, m by n and n by m, respectively, then D 2 X ¼ ½ðAW Þ A is the unique solution to the equations [2] ðAW Þkþ1 XW ¼ ðAW Þk ;
XWAWX ¼ X ;
AWX ¼ XWA;
where k ¼ IndðAW Þ, the index of AW [1], is the smallest nonnegative integer for which rank½ðAW Þk ¼ rank½ðAW Þkþ1 : The matrix X is called W -weighted Drazin inverse of A. We denote X by Ad;W . For B 2 C n n , there exits a unique matrix BD , the Drazin inverse of B such that BD BBD ¼ BD ;
BBD ¼ BD B;
Bkþ1 BD ¼ Bk ;
where k is the index of B. In particular, IndðBÞ ¼ 1, the Drazin inverse is called the group inverse and denoted by Ag . From Jordan canonical form theory, we get that for any complex m n matrix A with rankðAÞ ¼ r, there exits nonsingular matrices P and Q [18] such that A11 0 0 W11 1 A¼P ð1Þ Q ; W ¼Q P 1 ; 0 A22 0 W22 where P is a m m nonsingular matrix and Q is n n nonsingular matrix. Note that A11 , W11 are nonsingular submatrices. Now we can write the Ad;W of A in the form [18] ð1Þ 1 0 ðW 11 A11 W11 Þ Q1 : ð2Þ Ad;W ¼ P 0 0 Let the matrix M ¼ Ad;W WAW ;
N ¼ I M;
M ¼ WAWAd;W ;
N ¼ I M :
ð3Þ
Obviously M and N are oblique projections, as well as M and N , where ImðAÞ denotes the range of A and KerðAÞ is the null space of A. It is easy to check that
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419 k
ImðMÞ ¼ ImðAd;W Þ ¼ ImððAW Þ Þ ¼ KerðN Þ; ¼ ImðA d;W Þ;
ImðN Þ ¼ KerðM Þ;
405
ImðM Þ ¼ KerðN Þ ImðN Þ ¼ KerðAd;W Þ
k
¼ ðKerðWAÞ Þ:
ð4Þ
As for the recent results on the generalized inverse, we refer to [10–19].
2. Sylvester displacement rank Throughout the paper, U 2 C n n and V 2 C m m are some fixed matrices. The operator dðU ; V Þ ¼ AU VA is called the UV -displacement of A. To distinguish the displacement concept from the more general case in Section 3 we call it the Sylvester UV -displacement, since A is the solution of a certain Sylvester equation. We can easily find that for a nonsingular matrix A with UV -displacement structure the inverse matrix A1 possesses a VU -displacement structure, and the relation rankðA1 V UA1 Þ ¼ rankðAU VAÞ holds. We want to get an estimation for the VU -displacement rank of the generalized inverse of a matrix with displacement structure. First we investigate the displacement structure of Ad;W is the following representation. Proposition 2.1. Let A 2 C m n , U 2 C m m and V 2 C n n , and here Ad;W is the W weighted Drazin inverse of A. Then Ad;W V UAd;W ¼ Ad;W VN NUAd;W Ad;W ðWAWU VWAW ÞAd;W ;
ð5Þ
where M, N , M and N are defined in (3). Proof. The relation is immediately proved by the following equation: Ad;W ðWAWU VWAW ÞAd;W ¼ ðI N ÞUAd;W Ad;W V ðI N Þ:
ð6Þ
From (4), we obtain the following corollary. Corollary 2.1. The VU -displacement rank of Ad;W satisfies the following estimate: rankðAd;W V UAd;W Þ 6 rankðWAWU VWAW Þ þ rankðM VN Þ þ rankðNUMÞ:
ð7Þ
406
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
Proof. We will prove dim½ImðNUAd;W Þ ¼ rankðNUMÞ and rankðAd;W VN Þ ¼ rankðM VN Þ; rankðNUAd;W Þ ¼ dim½NU ImðAd;W Þ ¼ dim½ImðNUMÞ ¼ rankðNUMÞ;
ð8Þ
the last equality (8) can be proved similarly. Taking these into account, we obtain the above estimate.
h
Now we aim to obtain the estimation for the second and third terms on the right-hand side. Proposition 2.2. With the notation above, we get the estimate rankðM VN Þ þ rankðNUMÞ 6 rankðUG GV Þ;
ð9Þ
k
where G ¼ AðWAÞ . Proof. Set F ¼ UG GV , we get partition U11 U12 V11 P 1 UP ¼ ; Q1 VQ ¼ U21 U22 V21
V12 : V22
ð10Þ
From (10) we write the matrix F in the following form: P 1 FQ ¼ P 1 UPP 1 GQ P 1 GQQ1 VQ U11 U12 G11 G11 0 ¼ 0 0 0 U21 U22 U11 G11 G11 V11 G11 V12 : ¼ U21 G11 0 From [8] we know that U11 G11 G11 V11 rank U21 G11
G11 V12 0
0 0
V11 V21
V12 V22
P rankðG11 V12 Þ þ rankðU21 G11 Þ ¼ rankðV12 Þ þ rankðU21 Þ 0 V12 0 0 ¼ rank þ rank 0 0 U21 0 ¼ rankðM VN Þ þ rankðNUMÞ:
where G11 ¼ ðA11 W11 Þk A11 is nonsingular. Then the proof is over.
h
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407
From Propositions 2.1 and 2.2 we conclude the first main result. Theorem 2.1. For any A 2 C m n and Ad;W is its W -weighted Drazin inverse. Then rankðAd;W V UAd;w Þ 6 rankðWAWU VWAW Þ þ rankðUG GV Þ;
ð11Þ
where G ¼ AðWAÞk . Proposition 2.3. If WAWU ¼ UG, VWAW ¼ GV , then we get estimate rankðAd;w V UAd;w Þ 6 2 rankðWAWU VWAW Þ:
ð12Þ
3. Displacement structure for W-weighted Drazin inverse displacement In order to generalize Theorem 2.1 we introduce a generalized displacement 1 concept [4]. Let a ¼ ½aij 0 denote a nonsingular 2 · 2 matrix. We associate a with the polynomial in two variables aðk; lÞ ¼
1 X
aij ki lj
i;j¼0
and the linear fractional function fa ðkÞ ¼
a10 þ a11 k : a00 þ a01 k
ð13Þ
For any fixed U 2 C n n and V 2 C m m , the generalized aðV ; U Þ displacement of A 2 C m n generated by aðk; lÞ is defined by aðV ; U ÞA ¼
1 X
aij V i AU j :
i;j¼0
If a¼d
0 1 ; 1 0
we just get Sylvester displacement that we have discussed. If 1 0 a¼d ; 0 1 we get Stein displacement.
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M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419 1
1
1
1
Lemma 3.1 [4]. Let a ¼ ½aij 0 , b ¼ ½bij 0 , c ¼ ½cij 0 , d ¼ ½dij 0 be nonsingular 2 · 2 matrices such that a ¼ bT dc;
ð14Þ
then 1
ðb00 þ b01 kÞ aðk; lÞðc00 þ c01 lÞ
1
¼ dðfb ðkÞ; fc ðlÞÞ
ð15Þ
for all k, l with b00 þ b01 k 6¼ 0 and c00 þ c01 l 6¼ 0. 0 1 Lemma 3.2 [4]. Let d ¼ , then there exist 2 · 2 matrices b, c such that 1 0 (14) holds and b00 þ b01 V and c00 þ c01 U are invertible. Taking Lemmas 3.1 and 3.2 together, we obtain the following: Proposition 3.1 [4]. Let b and c be matrices satisfying the conditions in Lemma 3.2, then for A 2 C m n , aðV ; U ÞA ¼ ðb00 þ b01 V Þ½Afc ðU Þ fb ðV ÞAðc00 þ c01 U Þ: The following is very important to generalize Theorem 2.1 for general aðV ; U Þ displacement. Proposition 3.2 1 (a) If w ¼ ½wij 0 is nonsingular and w00 þ w01 V is invertible, then rankðM VN Þ ¼ rankðM Ve N Þ; where Ve fw ðV Þ is defined in (12). 1 (b) If / ¼ ½/ij 0 is nonsingular and /00 þ /01 U is invertible, then ~ MÞ; rankðNUMÞ ¼ rankðN U ~ ¼ f/ ðU Þ is defined in (12). where U Proof. We define S ¼ KerðGÞ \ KerðGV Þ; k
S1 ¼ KerðGÞ S;
where G ¼ AðWAÞ . We show that M VN is one-to-one on S1 . If M VN x ¼ 0 and x 2 S1 , then VN x 2 KerðM Þ ¼ KerðGÞ. That means GVN x ¼ GVx ¼ 0. Noting that x 2 KerðGÞ, we conclude x 2 S. Thus x ¼ 0. Furthermore, M VN vanishes on S. Since M VN x ¼ M Vx ¼ 0 for all x 2 S. Hence
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
rankðM VN Þ ¼ dimðS1 Þ:
409
ð16Þ
Analogously, we define f S; S 1 ¼ KerðGÞ f
f S ¼ KerðGÞ \ KerðG Ve Þ; and we will get S 1 Þ: rankðM Ve N Þ ¼ dimðf
ð17Þ
þw V bijectively maps S onto Now we show that the invertible matrix w 00 01 þw V Þx and ~ S. Suppose that x 2 S. Then x, Vx 2 KerðGÞ. Hence y ðw 10 11 e z ðw00 þ w01 V Þx are all contained in KerðGÞ. Thus y ¼ V z and we conclude that z, Vez 2 KerðGÞ, which implies z 2 S. Conversely, with the same arguments þw V Þ1 z 2 f we get ðw S for z 2 S. 00 01 This implies SÞ dimðS1 Þ ¼ dim½KerðGÞ dimðSÞ ¼ dim½KerðGÞ dimðf ¼ dimðf S 1 Þ: According to (16) and (17), we get assertion (a). Assertion (b) is proved analogously. h Now we can generalize Theorem 2.1 for general aðU ; V Þ displacement. Theorem 3.1. Let a; b be 2 · 2 nonsingular matrices, then rank½aðU ; V ÞAd;W 6 rank½aT ðU ; V ÞWAW þ rank½bðU ; V ÞG;
ð18Þ
k
where G ¼ AðWAÞ . Proof. According to Lemma 3.2 there exist 2 · 2 matrices w, x, y, z such that w00 þ w01 U , x00 þ x01 V , y00 þ y01 U , z00 þ z01 V are invertible and a ¼ wT dz; b ¼ xT dy: Hence, rank½aðU ; V ÞAd;W rank½aT ðV ; U ÞWAW ¼ rank½fw ðU ÞAd;W Ad;W fz ðV Þ rank½fz ðV ÞWAW WAWfw ðU Þ 6 rank½Nfw ðU ÞM þ rank½M fz ðV ÞN ¼ rank½Nfy ðU ÞM þ rank½M fx ðV ÞN 6 rank½fy ðU ÞG Gfx ðV Þ ¼ rank½bðU ; V ÞG:
410
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
4. Applications of the W-weighted Drazin inverse In this section we will present a selection for the application of Theorem 3.1. Through this section we only study the displacement structure of the W weighted Drazin inverse of A. Let Ad;W denote the W -weighted Drazin inverse of A. We study the displacement structure to close-to-Toeplitz matrices, close-toCauchy matrices, etc. 4.1. Close-to-Toeplitz matrices Let U , V be forward and backward matrices respectively. This means 0 1 0 B1 0 C C 2 C m m ; V ¼ Zn : U ¼ Zm B ð19Þ .. .. @ A . . 1
0
A matrix is said to be close-to-Toeplitz if it has UV -displacement structure for U ¼ Zm and V ¼ Zn . Toeplitz and Hankel sums, products, and the inverse of this matrices are close-to-Toeplitz. In Theorem 3.1 choosing 1 0 a¼b¼ : 0 1 We obtain the estimate rankðAd;W Zm Ad;W Zn Þ 6 rþ þ rankððAW Þk A Zm ðAW Þk AZn Þ:
ð20Þ
where rþ denotes the displacement ranks rþ ¼ rankðWAW Zn WAWZm Þ;
r ¼ rankðWAW Zn WAWZm Þ:
ð21Þ
Corollary 4.1.1 [3]. It is easy to check that if A is a Toeplitz matrix, i.e., A ¼ ðaij Þ, let W ¼ I, IndðAÞ ¼ k, then we can obtain rankðAD Zn AD Zn Þ 6 rankðA Zn AZn Þ þ rankðAkþ1 Zn AZn Þ:
ð22Þ
Corollary 4.1.2 [20]. If we let IndðAÞ ¼ 1, W ¼ I, we can get rankðAg Zn Ag Zn Þ 6 rþ þ rankðA2 Zn A2 Zn Þ 6 2 þ rankðA2 Zn A2 Zn Þ:
0
ð22 Þ
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411
4.2. Generalized Cauchy matrices In this section we study the generalized Cauchy matrices. Let U and V be diagonal matrices, n
U ¼ DðdÞ ¼ diagðdj Þ1 ;
n
V ¼ DðcÞ ¼ diagðci Þ1 :
A matrices A is called a generalized Cauchy matrices if and only if for certain c and d rank½ADðdÞ DðcÞA is small compared with the order of A. In case Ci 6¼ dj , for all i and j, generalized Cauchy matrices have the following form: n;n fi gj ; ð23Þ A¼ ci dj i¼1 j¼1 where fi , gj 2 C r and r ¼ rank½ADðdÞ DðcÞA. For r ¼ 1, f1 ¼ g1 ¼ 1, A is a classical Cauchy matrix. Another important case is the class of Loewner matrices
n;n ai bj : ð24Þ A¼ ci dj i¼1;j¼1 In this case the displacement rank r of A equals 2. We assume that ci dj ¼ di cj ;
i ¼ 1; . . . ; n; j ¼ 1; . . . ; n; and i 6¼ j:
ð25Þ
With the assumption (23) we know that rank½ADðcÞ DðdÞA ¼ rank½ADðdÞ DðcÞA:
ð26Þ
Corollary 4.2.1. It is easy to check by Theorem 2.1 that rankðAd;W DðcÞ DðdÞAd;W Þ 6 rankðWAWDðdÞ DðcÞWAW Þ k
A
þ rankððAW Þ ADðcÞ DðdÞðAW Þ Þ: 0
ð27 Þ
Corollary 4.2.2. A 2 C m n , is a generalized Cauchy matrix given by the form (23), then rank½Ag DðcÞ DðdÞAg 6 2r; where r ¼ rankðADðdÞ DðcÞAÞ.
ð27Þ
412
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
5. The perturbation formula for the W-weighted Drazin inverse One of the most powerful methods is through the Jordan canonical form theory. Now we shall use this theory to obtain the expression of the perturbation of the W -weighted inverse. Let Ap denote the eigenprojection of a square matrix to the eigenvalue 0. p
Lemma 5.1. Let A 2 C m n , W 2 C n m and G 2 C n n . Then G ¼ ðWAÞ if and only if G2 ¼ G, WAG ¼ GWA, rðWAGÞ ¼ 0, WA þ G is nonsingular. We can get 0 0 p ðWAÞ ¼ I WAWAd;W ¼ Q Q1 : ð28Þ 0 I Next we will discuss the properties of matrices with equal eigenprojections at zero. Theorem 5.1. Let A,B 2 C m n , W 2 C n m , the following conditions on WB are equivalent: p
p
ðWBÞ ¼ ðWAÞ ; p p p p ðWAÞ WB ¼ WBðWAÞ , and WBðWAÞ is nilpotent, WB þ ðWAÞ is nonsingular; B11 is invertible, W22 B22 is nilpotent, and Wii Bij ¼ 0, for i 6¼ j; I þ Ad;W W ðB AÞW is nonsingular, ðWAÞp WB ¼ WBðWAÞp , and WBðWAÞp is nilpotent; (5) Bd;W ¼ ðI þ Ad;W W ðB AÞW Þ1 Ad;W ; (6) Bd;W Ad;W ¼ Ad;W W ðB AÞWBd;W .
(1) (2) (3) (4)
p
p
Proof. We observe that ðWAÞ WB ¼ WBðWAÞ implies W11 B11 0 WB ¼ Q Q1 : 0 W22 B22 Q1 0 0 In fact, from ðWAÞp ¼ Q we can get 0 I 0 0 0 Q Q1 ¼ ðWAÞp WB ¼ WBðWAÞp ¼ Q W22 B21 W22 B22 0 So, W22 B21 ¼ W11 B12 ¼ 0: From the lemma, we have (1) () (2). (2) ) (3): Since WB þ ðWAÞp is nonsingular, and
W11 B12 Q1 : W22 B22
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
W11 B12 p WB þ ðWAÞ ¼ Q 0
0 I þ W22 B22
413
Q1 ;
then W11 B11 is nonsingular and W11 ; B11 is nonsingular. p Since WBðWAÞ is nilpotent, and 0 0 WBðWAÞp ¼ Q Q1 ; 0 W22 B22 so W22 B22 is nilpotent. (3) ) (4): I þ Ad;W W ðB AÞW ¼ P
ðW11 A11 W11 Þ1 W11 B11 W11 0
0 P 1 I
since W11 , B11 is nonsingular, then I þ Ad;W W ðB AÞW is nonsingular. The rest is verified easily. (4) ) (5): Since I þ Ad;W W ðB AÞW is nonsingular, and B11 is nonsingular. Q1 0 0 p WBðWAÞ ¼ Q 0 W22 B22 is nilpotent and W22 B22 is nilpotent. So, ðI þ Ad;W W ðB AÞW ÞBd;W ¼P ¼P
ðW11 A11 W11 Þ1 W11 B11 W11
0
0
I
ðW11 A11 W11 Þ 0
1
!
ðW11 B11 W11 Þ1
0
0
0
! 0 Q1 ¼ Ad;W 0
! Q1
(5) ) (6): It is verified by a calculation. (6) ) (1): From Ad;W ¼ ðI Ad;W W ðABÞW ÞBd;W , we have N ðWBd;W Þ N ðWAd;W Þ; from Bd;w ¼ Ad;W ðI þ W ðA BÞWBd;W Þ, we can get RðWBd;W Þ RðWAd;W Þ: So, d ¼ nullityðWBd;W Þ þ rankðWBd;W Þ 6 nullityðWAd;W Þ þ rankðWAd;W Þ ¼ d;
414
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419 p
then N ðWBd;w Þ ¼ N ðWAd;W Þ, RðWBd;W Þ ¼ RðWAd;W Þ. We can get ðWBÞ ¼ p ðWAÞ . h Proposition 5.1. The conditions (4)–(6) of the preceding theorem have the following equivalent versions: p
p
p
(4) I þ W ðB AÞWAd;W is nonsingular, ðWAÞ WB ¼ WBðWAÞ and WBðWAÞ is nilpotent, (5) Bd;W ¼ Ad;W ðI þ W ðB AÞWAd;W Þ1 , (6) Bd;W ¼ Ad;W ¼ Bd;W W ðB AÞWAd;W . Theorem 5.2. Let A, B 2 C m n , W 2 C n m , and ðWBÞp ¼ ðWAÞp . Then, kAd;W k 6 kBd;W k; 1 þ kAd;W W ðB AÞW k
ð29Þ
if kAd;W W ðB AÞW k < 1, then kAd;W k 1 kAd;W W ðB AÞW k
ð30Þ
kBd;W Ad;W k kAd;W ðB AÞW k : 6 Ad;W 1 kAd;W W ðB AÞW k
ð31Þ
kBd;W k 6 and
Proof. By the above theorem, Bd;W Ad;W ¼ Ad;W W ðB AÞWBd;W , then Ad;W ¼ Bd;W Ad;W W ðB AÞWBd;W : Applying the norm to both sides, kAd;W k 6 kBd;W k þ kAd;W W ðB AÞkkBd;W k to obtain kAd;W k 6 kBd;W k: 1 þ kAd;W W ðB AÞW k Let kAd;W W ðB AÞW k < 1, since Bd;W ¼ Ad;W þ Ad;W W ðB AÞWBd;W , then kBd;W k 6 kAd;W k þ kAd;W W ðB AÞW kkBd;W k i.e., kBd;W k 6
kAd;W k : 1 kAd;W W ðB AÞW k
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
415
Finally, since Bd;W Ad;W ¼ Ad;W W ðB AÞW ðAd;W þ Bd;W Ad;W Þ kBd;W Ad;W k 6 kAd;W W ðB AÞW kðkAd;W k þ kBd;W Ad;W kÞ; we get kBd;W k kAd;W k kAd;W W ðB AÞW k 6 : kAd;W k 1 kAd;W W ðB AÞW k
p
p
Proposition 5.2. The condition ðWBÞ ¼ ðWAÞ could be placed with any of the equivalent formulations discussed above. Proposition 5.3. If kAd;W kkW ðB AÞW k < 1 in the proceeding theorem, then (31) becomes kBd;W Ad;W k kAd;W W ðB AÞW k jd;W ðAÞD 6 ; 6 Ad;W 1 kAd;W W ðB AÞW k 1 jd;W ðAÞD ðBAÞW k where jd;W ðAÞ ¼ kAd;W kkWAW k, D ¼ kWkWAW . k
Theorem 5.3. Let B ¼ A þ E 2 C m n , WE ¼ WEðWAÞðWAÞ ðAW ÞEW . Then
D
D
and EW ¼ ðAW Þ
1
Bd;w ¼ ½I þ Ad;W WEW Ad;W :
Proof. We define E11 P 1 EQ ¼ E21 thus, we obtain E11 E¼P E21
E12 ; E22
ð32Þ
E12 Q1 : E22
ð33Þ
So we can apply the matrix E in the above form to the condition WE ¼ WEðWAÞðWAÞD and EW ¼ ðAW ÞD ðAW ÞEW , we can get W11 E11 ¼ W22 E22 ¼ 0 and E21 W11 ¼ E22 W22 ¼ 0 that is E12 ¼ 0, E21 ¼ 0. E11 W11 0 E11 0 P 1 ; E¼P Q1 and EW ¼ P 0 E22 0 0
416
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
Bd;W ¼ ½ðBW ÞD 2 B ¼ ½ðAW þ EW ÞD 2 ðA þ EÞ (
D )2 0 A11 W11 þ E11 W11 1 ¼ P P 0 A22 W22 0 A11 þ E11 P Q1 0 A22 þ E22 ! 1 W111 ðA11 þ E11 Þ11 0 Q1 ¼P 0 0 ¼P ¼P
! 1 1 1 ½I þ ðW11 A11 W11 Þ W11 E11 W11 ðW11 A11 W11 Þ 0 Q1 0 0 ! ! ½I þ ðW11 A11 W11 Þ1 W11 E11 W11 1 0 ðW11 A11 W11 Þ1 0 1 P P Q1 0 I 0 0
¼ ½I þ Ad;W WEW 1 Ad;W :
From Theorem 5.3 now it is not difficult to get the following two corollaries: D
Corollary 5.1. Let B ¼ A þ E 2 C m n and EW ¼ ðAW Þ ðAW ÞEW with k ¼ IndðAÞ. Then Bd;W ¼ Ad;W ½I þ WEWAd;W WAW 1 fI þ WAd;W WAW ½Ad;W ðI þ WEWAd;W Þ1 WAWAd;W WEðI WAWAd;W Þg þ
k1 X
1
Ad;W ðI þ WEWAd;W WAW Þ
i¼0
ðBWAd;W WAW Þ
ðiþ2Þ
i
EW ðI Ad;W WAW Þ½ðI Ad;W WAW Þ B
ðI WAWAd;W Þ:
Proof. The proof is analogous to the proof of Theorem 5.3. Since D EW ¼ ðAW Þ ðAW ÞEW , we can get E21 ¼ 0. E11 W11 E12 E22 E11 E12 P 1 ; E¼P Q1 and EW ¼ P 0 E22 0 0 Bd;W ¼ ½ðBW ÞD 2 B ¼ ½ðAW þ EW ÞD 2 ðA þ EÞ (
D )2 A11 W11 þ E11 W11 E12 E22 A11 þ E11 1 ¼ P P P 0 A22 W22 0
E12 A22 þ E22
Q1
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
W111 ðA11 þ E11 Þ
¼P
1
X
!2
A11 þ E11
E12
417
Q1
A22 þ E22 ! L LðA11 þ E11 Þ E12 þ LW11 X ðA22 þ E22 Þ Q1 ¼P 0 0 ! 1 L 0 þ E Þ E þ W X ðA þ E Þ I ðA 11 11 12 11 22 22 ¼P Q1 Q Q1 ; 0 0 0 0 0
0
0
1
where 1
L ¼ ½W11 ðA11 þ E11 W11 Þ ; and X ¼
l1 X ½ðA11 W11 þ E11 W11 Þ1 2 E12 W22 ðA22 W22 Þi : i¼0
As the similar proof of the Theorem 5.3, we can obtain L 0 1 P Q1 ¼ Ad;W ½I þ WEWAd;W WAW : 0 0 So it is not different to get the result: 1
Bd;W ¼ Ad;W ½I þ WEWAd;W WAW fI þ WAd;W WAW ½Ad;W ðI þ WEWAd;W Þ WAWAd;W WEðI WAWAd;W Þg þ
l1 X
1
Ad;W ðI þ WEWAd;W WAW Þ1
i¼0 ðiþ2Þ
ðBWAd;W WAW Þ ðI WAWAd;W Þ:
i
EW ðI Ad;W WAW Þ½AW ðI Ad;W WAW Þ B
Corollary 5.2. Let B ¼ A þ E 2 C m n , and WE ¼ WEðWAÞðWAÞ IngðAÞ. Then ( Bd;W ¼
D
with k ¼
I þ ½ðI Ad;W WAW ÞEWAd;W WAWAd;W ðI þ WEWAd;W WAW Þ1 þ
l1 X
BðI WAWAd;W Þ½WAðI WAWAd;W Þi WEðI WAWAd;W Þ
i¼0
ðWBWAWAd;W Þ
) ðiþ2Þ
WAWAd;W
1
Ad;W ðI þ WEWAd;W WAW Þ :
418
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
6. Concluding remarks In this paper we study the displacement structure and the perturbation of the W -weighted Drazin inverse of a complex matrix. It is natural to ask if we can extend our results to linear operators in Hilbert space. This will be the future research.
Acknowledgements This project is supported by National Natural Science Foundation of China under grant 19901006 and 10171021.
References [1] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, Wiley, New York, 1974, 2nd ed., Springer-Verlag, New York, 2003. [2] R.E. Cline, T.N.E. Greville, A Drazin inverse for rectangular matrices, Linear Algebra Appl. 29 (1980) 53–62. [3] H. Diao, Y. Wei, S. Qiao, Displacement rank of the Drazin inverse, J. Comput. Appl. Math., in press. [4] G. Heinig, F. Hellinger, Displacement structure of pseudoinverses, Linear Algebra Appl. 197– 198 (1994) 623–649. [5] G. Heinig, F. Hellinger, Displacement structure of generalized inverse matrices, Linear Algebra Appl. 211 (1994) 67–83. [6] T. Kailath, S.Y. Kung, M. Morf, Displacement rank of matrices and linear equations, J. Math. Anal. Appl. 68 (1979) 395–407. [7] T. Kailath, A. Sayed, Displacement structure: theory and applications, SIAM Rev. 37 (1995) 297–386. [8] G. Marsaglia, G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra 2 (1974) 269–292. [9] V.Y. Pan, Structured Matrices and Polynomials: Unified Superfast Algorithms, Birkhauser– Springer, 2001. [10] V. Rakocevic, Y. Wei, The perturbation theory for the Drazin inverse and its applications II, J. Aust. Math. Soc. 70 (2001) 189–197. [11] V. Rakocevic, Y. Wei, A weighted Drazin inverse and applications, Linear Algebra Appl. 350 (2002) 25–39. [12] V. Rakocevic, Y. Wei, The representation and approximation of the W -weighted Drazin inverse of linear operators in Hilbert space, Appl. Math. Comput. 141 (2003) 455– 470. [13] Y. Wei, G. Wang, The perturbation theory for the Drazin inverse and its applications, Linear Algebra Appl. 258 (1997) 179–186. [14] Y. Wei, On the perturbation of the group inverse and oblique projection, Appl. Math. Comput. 98 (1999) 29–42. [15] Y. Wei, The Drazin inverse of updating of a square matrix with application to perturbation formula, Appl. Math. Comput. 108 (2000) 77–83. [16] Y. Wei, Perturbation bound of the Drazin inverse, Appl. Math. Comput. 125 (2002) 231–244.
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[17] Y. Wei, A characterization for the W -weighted Drazin inverse and a Cramer rule for the W weighted Drazin inverse solution, Appl. Math. Comput. 125 (2002) 303–310. [18] Y. Wei, Integral representation of the W -weighted Drazin inverse, Appl. Math. Comput. 144 (2003) 3–10. [19] Y. Wei, C. Woo, T. Lei, A note on perturbation of the W -weighted Drazin inverse, Appl. Math. Comput. 149 (2004) 423–430. [20] Y. Wei, M. Ng, Displacement structure of group inverses, Number. Linear Algebra Appl., in press.
Displacement structure of W -weighted Drazin inverse Ad;W and its perturbation Mei Qin *, Guanglin Wang Institute of Mathematics, Fudan University, Shanghai 200433, PR China
Abstract It is well known that matrices with a UV -displacement structure possess generalized inverse with a VU-displacement structure. Estimations for the displacement rank of Ad;W U VAd;W are presented, where Ad;W is the W -weighted Drazin inverse of A. We extend the results for the Drazin inverse and group inverse, respectively. In the end, the perturbation of W -weighted Drazin inverse Ad;W is studied. 2004 Elsevier Inc. All rights reserved. Keywords: Displacement; W -weighted Drazin inverse; Perturbation
1. Introduction and preliminaries The present paper is to study the generalized inverse of a structured matrix. To begin with we recall some facts concerning the regular inversion of structured matrices, which are motivated by [6,7]. If the rank of a matrix’s displacement is small, fast algorithms for the matrix are available. A matrix is called matrices with displacement structure [4,5,9] if and only if the rank of the matrix AU VA or A VAU is small compared with the order of the matrix A. The rank of AU VA is said to be Sylvester UV -displacement rank and the rank of A VAU is called the Stein UV -displacement rank of A, since A is the solution of a Sylvester or Stein equations, respectively.
*
Corresponding author. E-mail address: [email protected] (M. Qin).
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.140
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M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
As we all know that fast inversion algorithms for matrix A can be constructed if A is matrix with displacement structure. We are interested in the generalized inverse with as small as possible a displacement rank. We present the estimate for the rank of Ad;W U VAd;W , where Ad;W is the W -weighted Drazin inverse of A. Then we give an explicit estimation for general displacement. Our results cover the previous result [3,4,20] for the Drazin inverse and group inverse, respectively. If A and W are any complex matrices, m by n and n by m, respectively, then D 2 X ¼ ½ðAW Þ A is the unique solution to the equations [2] ðAW Þkþ1 XW ¼ ðAW Þk ;
XWAWX ¼ X ;
AWX ¼ XWA;
where k ¼ IndðAW Þ, the index of AW [1], is the smallest nonnegative integer for which rank½ðAW Þk ¼ rank½ðAW Þkþ1 : The matrix X is called W -weighted Drazin inverse of A. We denote X by Ad;W . For B 2 C n n , there exits a unique matrix BD , the Drazin inverse of B such that BD BBD ¼ BD ;
BBD ¼ BD B;
Bkþ1 BD ¼ Bk ;
where k is the index of B. In particular, IndðBÞ ¼ 1, the Drazin inverse is called the group inverse and denoted by Ag . From Jordan canonical form theory, we get that for any complex m n matrix A with rankðAÞ ¼ r, there exits nonsingular matrices P and Q [18] such that A11 0 0 W11 1 A¼P ð1Þ Q ; W ¼Q P 1 ; 0 A22 0 W22 where P is a m m nonsingular matrix and Q is n n nonsingular matrix. Note that A11 , W11 are nonsingular submatrices. Now we can write the Ad;W of A in the form [18] ð1Þ 1 0 ðW 11 A11 W11 Þ Q1 : ð2Þ Ad;W ¼ P 0 0 Let the matrix M ¼ Ad;W WAW ;
N ¼ I M;
M ¼ WAWAd;W ;
N ¼ I M :
ð3Þ
Obviously M and N are oblique projections, as well as M and N , where ImðAÞ denotes the range of A and KerðAÞ is the null space of A. It is easy to check that
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419 k
ImðMÞ ¼ ImðAd;W Þ ¼ ImððAW Þ Þ ¼ KerðN Þ; ¼ ImðA d;W Þ;
ImðN Þ ¼ KerðM Þ;
405
ImðM Þ ¼ KerðN Þ ImðN Þ ¼ KerðAd;W Þ
k
¼ ðKerðWAÞ Þ:
ð4Þ
As for the recent results on the generalized inverse, we refer to [10–19].
2. Sylvester displacement rank Throughout the paper, U 2 C n n and V 2 C m m are some fixed matrices. The operator dðU ; V Þ ¼ AU VA is called the UV -displacement of A. To distinguish the displacement concept from the more general case in Section 3 we call it the Sylvester UV -displacement, since A is the solution of a certain Sylvester equation. We can easily find that for a nonsingular matrix A with UV -displacement structure the inverse matrix A1 possesses a VU -displacement structure, and the relation rankðA1 V UA1 Þ ¼ rankðAU VAÞ holds. We want to get an estimation for the VU -displacement rank of the generalized inverse of a matrix with displacement structure. First we investigate the displacement structure of Ad;W is the following representation. Proposition 2.1. Let A 2 C m n , U 2 C m m and V 2 C n n , and here Ad;W is the W weighted Drazin inverse of A. Then Ad;W V UAd;W ¼ Ad;W VN NUAd;W Ad;W ðWAWU VWAW ÞAd;W ;
ð5Þ
where M, N , M and N are defined in (3). Proof. The relation is immediately proved by the following equation: Ad;W ðWAWU VWAW ÞAd;W ¼ ðI N ÞUAd;W Ad;W V ðI N Þ:
ð6Þ
From (4), we obtain the following corollary. Corollary 2.1. The VU -displacement rank of Ad;W satisfies the following estimate: rankðAd;W V UAd;W Þ 6 rankðWAWU VWAW Þ þ rankðM VN Þ þ rankðNUMÞ:
ð7Þ
406
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
Proof. We will prove dim½ImðNUAd;W Þ ¼ rankðNUMÞ and rankðAd;W VN Þ ¼ rankðM VN Þ; rankðNUAd;W Þ ¼ dim½NU ImðAd;W Þ ¼ dim½ImðNUMÞ ¼ rankðNUMÞ;
ð8Þ
the last equality (8) can be proved similarly. Taking these into account, we obtain the above estimate.
h
Now we aim to obtain the estimation for the second and third terms on the right-hand side. Proposition 2.2. With the notation above, we get the estimate rankðM VN Þ þ rankðNUMÞ 6 rankðUG GV Þ;
ð9Þ
k
where G ¼ AðWAÞ . Proof. Set F ¼ UG GV , we get partition U11 U12 V11 P 1 UP ¼ ; Q1 VQ ¼ U21 U22 V21
V12 : V22
ð10Þ
From (10) we write the matrix F in the following form: P 1 FQ ¼ P 1 UPP 1 GQ P 1 GQQ1 VQ U11 U12 G11 G11 0 ¼ 0 0 0 U21 U22 U11 G11 G11 V11 G11 V12 : ¼ U21 G11 0 From [8] we know that U11 G11 G11 V11 rank U21 G11
G11 V12 0
0 0
V11 V21
V12 V22
P rankðG11 V12 Þ þ rankðU21 G11 Þ ¼ rankðV12 Þ þ rankðU21 Þ 0 V12 0 0 ¼ rank þ rank 0 0 U21 0 ¼ rankðM VN Þ þ rankðNUMÞ:
where G11 ¼ ðA11 W11 Þk A11 is nonsingular. Then the proof is over.
h
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
407
From Propositions 2.1 and 2.2 we conclude the first main result. Theorem 2.1. For any A 2 C m n and Ad;W is its W -weighted Drazin inverse. Then rankðAd;W V UAd;w Þ 6 rankðWAWU VWAW Þ þ rankðUG GV Þ;
ð11Þ
where G ¼ AðWAÞk . Proposition 2.3. If WAWU ¼ UG, VWAW ¼ GV , then we get estimate rankðAd;w V UAd;w Þ 6 2 rankðWAWU VWAW Þ:
ð12Þ
3. Displacement structure for W-weighted Drazin inverse displacement In order to generalize Theorem 2.1 we introduce a generalized displacement 1 concept [4]. Let a ¼ ½aij 0 denote a nonsingular 2 · 2 matrix. We associate a with the polynomial in two variables aðk; lÞ ¼
1 X
aij ki lj
i;j¼0
and the linear fractional function fa ðkÞ ¼
a10 þ a11 k : a00 þ a01 k
ð13Þ
For any fixed U 2 C n n and V 2 C m m , the generalized aðV ; U Þ displacement of A 2 C m n generated by aðk; lÞ is defined by aðV ; U ÞA ¼
1 X
aij V i AU j :
i;j¼0
If a¼d
0 1 ; 1 0
we just get Sylvester displacement that we have discussed. If 1 0 a¼d ; 0 1 we get Stein displacement.
408
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419 1
1
1
1
Lemma 3.1 [4]. Let a ¼ ½aij 0 , b ¼ ½bij 0 , c ¼ ½cij 0 , d ¼ ½dij 0 be nonsingular 2 · 2 matrices such that a ¼ bT dc;
ð14Þ
then 1
ðb00 þ b01 kÞ aðk; lÞðc00 þ c01 lÞ
1
¼ dðfb ðkÞ; fc ðlÞÞ
ð15Þ
for all k, l with b00 þ b01 k 6¼ 0 and c00 þ c01 l 6¼ 0. 0 1 Lemma 3.2 [4]. Let d ¼ , then there exist 2 · 2 matrices b, c such that 1 0 (14) holds and b00 þ b01 V and c00 þ c01 U are invertible. Taking Lemmas 3.1 and 3.2 together, we obtain the following: Proposition 3.1 [4]. Let b and c be matrices satisfying the conditions in Lemma 3.2, then for A 2 C m n , aðV ; U ÞA ¼ ðb00 þ b01 V Þ½Afc ðU Þ fb ðV ÞAðc00 þ c01 U Þ: The following is very important to generalize Theorem 2.1 for general aðV ; U Þ displacement. Proposition 3.2 1 (a) If w ¼ ½wij 0 is nonsingular and w00 þ w01 V is invertible, then rankðM VN Þ ¼ rankðM Ve N Þ; where Ve fw ðV Þ is defined in (12). 1 (b) If / ¼ ½/ij 0 is nonsingular and /00 þ /01 U is invertible, then ~ MÞ; rankðNUMÞ ¼ rankðN U ~ ¼ f/ ðU Þ is defined in (12). where U Proof. We define S ¼ KerðGÞ \ KerðGV Þ; k
S1 ¼ KerðGÞ S;
where G ¼ AðWAÞ . We show that M VN is one-to-one on S1 . If M VN x ¼ 0 and x 2 S1 , then VN x 2 KerðM Þ ¼ KerðGÞ. That means GVN x ¼ GVx ¼ 0. Noting that x 2 KerðGÞ, we conclude x 2 S. Thus x ¼ 0. Furthermore, M VN vanishes on S. Since M VN x ¼ M Vx ¼ 0 for all x 2 S. Hence
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
rankðM VN Þ ¼ dimðS1 Þ:
409
ð16Þ
Analogously, we define f S; S 1 ¼ KerðGÞ f
f S ¼ KerðGÞ \ KerðG Ve Þ; and we will get S 1 Þ: rankðM Ve N Þ ¼ dimðf
ð17Þ
þw V bijectively maps S onto Now we show that the invertible matrix w 00 01 þw V Þx and ~ S. Suppose that x 2 S. Then x, Vx 2 KerðGÞ. Hence y ðw 10 11 e z ðw00 þ w01 V Þx are all contained in KerðGÞ. Thus y ¼ V z and we conclude that z, Vez 2 KerðGÞ, which implies z 2 S. Conversely, with the same arguments þw V Þ1 z 2 f we get ðw S for z 2 S. 00 01 This implies SÞ dimðS1 Þ ¼ dim½KerðGÞ dimðSÞ ¼ dim½KerðGÞ dimðf ¼ dimðf S 1 Þ: According to (16) and (17), we get assertion (a). Assertion (b) is proved analogously. h Now we can generalize Theorem 2.1 for general aðU ; V Þ displacement. Theorem 3.1. Let a; b be 2 · 2 nonsingular matrices, then rank½aðU ; V ÞAd;W 6 rank½aT ðU ; V ÞWAW þ rank½bðU ; V ÞG;
ð18Þ
k
where G ¼ AðWAÞ . Proof. According to Lemma 3.2 there exist 2 · 2 matrices w, x, y, z such that w00 þ w01 U , x00 þ x01 V , y00 þ y01 U , z00 þ z01 V are invertible and a ¼ wT dz; b ¼ xT dy: Hence, rank½aðU ; V ÞAd;W rank½aT ðV ; U ÞWAW ¼ rank½fw ðU ÞAd;W Ad;W fz ðV Þ rank½fz ðV ÞWAW WAWfw ðU Þ 6 rank½Nfw ðU ÞM þ rank½M fz ðV ÞN ¼ rank½Nfy ðU ÞM þ rank½M fx ðV ÞN 6 rank½fy ðU ÞG Gfx ðV Þ ¼ rank½bðU ; V ÞG:
410
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
4. Applications of the W-weighted Drazin inverse In this section we will present a selection for the application of Theorem 3.1. Through this section we only study the displacement structure of the W weighted Drazin inverse of A. Let Ad;W denote the W -weighted Drazin inverse of A. We study the displacement structure to close-to-Toeplitz matrices, close-toCauchy matrices, etc. 4.1. Close-to-Toeplitz matrices Let U , V be forward and backward matrices respectively. This means 0 1 0 B1 0 C C 2 C m m ; V ¼ Zn : U ¼ Zm B ð19Þ .. .. @ A . . 1
0
A matrix is said to be close-to-Toeplitz if it has UV -displacement structure for U ¼ Zm and V ¼ Zn . Toeplitz and Hankel sums, products, and the inverse of this matrices are close-to-Toeplitz. In Theorem 3.1 choosing 1 0 a¼b¼ : 0 1 We obtain the estimate rankðAd;W Zm Ad;W Zn Þ 6 rþ þ rankððAW Þk A Zm ðAW Þk AZn Þ:
ð20Þ
where rþ denotes the displacement ranks rþ ¼ rankðWAW Zn WAWZm Þ;
r ¼ rankðWAW Zn WAWZm Þ:
ð21Þ
Corollary 4.1.1 [3]. It is easy to check that if A is a Toeplitz matrix, i.e., A ¼ ðaij Þ, let W ¼ I, IndðAÞ ¼ k, then we can obtain rankðAD Zn AD Zn Þ 6 rankðA Zn AZn Þ þ rankðAkþ1 Zn AZn Þ:
ð22Þ
Corollary 4.1.2 [20]. If we let IndðAÞ ¼ 1, W ¼ I, we can get rankðAg Zn Ag Zn Þ 6 rþ þ rankðA2 Zn A2 Zn Þ 6 2 þ rankðA2 Zn A2 Zn Þ:
0
ð22 Þ
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
411
4.2. Generalized Cauchy matrices In this section we study the generalized Cauchy matrices. Let U and V be diagonal matrices, n
U ¼ DðdÞ ¼ diagðdj Þ1 ;
n
V ¼ DðcÞ ¼ diagðci Þ1 :
A matrices A is called a generalized Cauchy matrices if and only if for certain c and d rank½ADðdÞ DðcÞA is small compared with the order of A. In case Ci 6¼ dj , for all i and j, generalized Cauchy matrices have the following form: n;n fi gj ; ð23Þ A¼ ci dj i¼1 j¼1 where fi , gj 2 C r and r ¼ rank½ADðdÞ DðcÞA. For r ¼ 1, f1 ¼ g1 ¼ 1, A is a classical Cauchy matrix. Another important case is the class of Loewner matrices
n;n ai bj : ð24Þ A¼ ci dj i¼1;j¼1 In this case the displacement rank r of A equals 2. We assume that ci dj ¼ di cj ;
i ¼ 1; . . . ; n; j ¼ 1; . . . ; n; and i 6¼ j:
ð25Þ
With the assumption (23) we know that rank½ADðcÞ DðdÞA ¼ rank½ADðdÞ DðcÞA:
ð26Þ
Corollary 4.2.1. It is easy to check by Theorem 2.1 that rankðAd;W DðcÞ DðdÞAd;W Þ 6 rankðWAWDðdÞ DðcÞWAW Þ k
A
þ rankððAW Þ ADðcÞ DðdÞðAW Þ Þ: 0
ð27 Þ
Corollary 4.2.2. A 2 C m n , is a generalized Cauchy matrix given by the form (23), then rank½Ag DðcÞ DðdÞAg 6 2r; where r ¼ rankðADðdÞ DðcÞAÞ.
ð27Þ
412
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
5. The perturbation formula for the W-weighted Drazin inverse One of the most powerful methods is through the Jordan canonical form theory. Now we shall use this theory to obtain the expression of the perturbation of the W -weighted inverse. Let Ap denote the eigenprojection of a square matrix to the eigenvalue 0. p
Lemma 5.1. Let A 2 C m n , W 2 C n m and G 2 C n n . Then G ¼ ðWAÞ if and only if G2 ¼ G, WAG ¼ GWA, rðWAGÞ ¼ 0, WA þ G is nonsingular. We can get 0 0 p ðWAÞ ¼ I WAWAd;W ¼ Q Q1 : ð28Þ 0 I Next we will discuss the properties of matrices with equal eigenprojections at zero. Theorem 5.1. Let A,B 2 C m n , W 2 C n m , the following conditions on WB are equivalent: p
p
ðWBÞ ¼ ðWAÞ ; p p p p ðWAÞ WB ¼ WBðWAÞ , and WBðWAÞ is nilpotent, WB þ ðWAÞ is nonsingular; B11 is invertible, W22 B22 is nilpotent, and Wii Bij ¼ 0, for i 6¼ j; I þ Ad;W W ðB AÞW is nonsingular, ðWAÞp WB ¼ WBðWAÞp , and WBðWAÞp is nilpotent; (5) Bd;W ¼ ðI þ Ad;W W ðB AÞW Þ1 Ad;W ; (6) Bd;W Ad;W ¼ Ad;W W ðB AÞWBd;W .
(1) (2) (3) (4)
p
p
Proof. We observe that ðWAÞ WB ¼ WBðWAÞ implies W11 B11 0 WB ¼ Q Q1 : 0 W22 B22 Q1 0 0 In fact, from ðWAÞp ¼ Q we can get 0 I 0 0 0 Q Q1 ¼ ðWAÞp WB ¼ WBðWAÞp ¼ Q W22 B21 W22 B22 0 So, W22 B21 ¼ W11 B12 ¼ 0: From the lemma, we have (1) () (2). (2) ) (3): Since WB þ ðWAÞp is nonsingular, and
W11 B12 Q1 : W22 B22
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
W11 B12 p WB þ ðWAÞ ¼ Q 0
0 I þ W22 B22
413
Q1 ;
then W11 B11 is nonsingular and W11 ; B11 is nonsingular. p Since WBðWAÞ is nilpotent, and 0 0 WBðWAÞp ¼ Q Q1 ; 0 W22 B22 so W22 B22 is nilpotent. (3) ) (4): I þ Ad;W W ðB AÞW ¼ P
ðW11 A11 W11 Þ1 W11 B11 W11 0
0 P 1 I
since W11 , B11 is nonsingular, then I þ Ad;W W ðB AÞW is nonsingular. The rest is verified easily. (4) ) (5): Since I þ Ad;W W ðB AÞW is nonsingular, and B11 is nonsingular. Q1 0 0 p WBðWAÞ ¼ Q 0 W22 B22 is nilpotent and W22 B22 is nilpotent. So, ðI þ Ad;W W ðB AÞW ÞBd;W ¼P ¼P
ðW11 A11 W11 Þ1 W11 B11 W11
0
0
I
ðW11 A11 W11 Þ 0
1
!
ðW11 B11 W11 Þ1
0
0
0
! 0 Q1 ¼ Ad;W 0
! Q1
(5) ) (6): It is verified by a calculation. (6) ) (1): From Ad;W ¼ ðI Ad;W W ðABÞW ÞBd;W , we have N ðWBd;W Þ N ðWAd;W Þ; from Bd;w ¼ Ad;W ðI þ W ðA BÞWBd;W Þ, we can get RðWBd;W Þ RðWAd;W Þ: So, d ¼ nullityðWBd;W Þ þ rankðWBd;W Þ 6 nullityðWAd;W Þ þ rankðWAd;W Þ ¼ d;
414
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419 p
then N ðWBd;w Þ ¼ N ðWAd;W Þ, RðWBd;W Þ ¼ RðWAd;W Þ. We can get ðWBÞ ¼ p ðWAÞ . h Proposition 5.1. The conditions (4)–(6) of the preceding theorem have the following equivalent versions: p
p
p
(4) I þ W ðB AÞWAd;W is nonsingular, ðWAÞ WB ¼ WBðWAÞ and WBðWAÞ is nilpotent, (5) Bd;W ¼ Ad;W ðI þ W ðB AÞWAd;W Þ1 , (6) Bd;W ¼ Ad;W ¼ Bd;W W ðB AÞWAd;W . Theorem 5.2. Let A, B 2 C m n , W 2 C n m , and ðWBÞp ¼ ðWAÞp . Then, kAd;W k 6 kBd;W k; 1 þ kAd;W W ðB AÞW k
ð29Þ
if kAd;W W ðB AÞW k < 1, then kAd;W k 1 kAd;W W ðB AÞW k
ð30Þ
kBd;W Ad;W k kAd;W ðB AÞW k : 6 Ad;W 1 kAd;W W ðB AÞW k
ð31Þ
kBd;W k 6 and
Proof. By the above theorem, Bd;W Ad;W ¼ Ad;W W ðB AÞWBd;W , then Ad;W ¼ Bd;W Ad;W W ðB AÞWBd;W : Applying the norm to both sides, kAd;W k 6 kBd;W k þ kAd;W W ðB AÞkkBd;W k to obtain kAd;W k 6 kBd;W k: 1 þ kAd;W W ðB AÞW k Let kAd;W W ðB AÞW k < 1, since Bd;W ¼ Ad;W þ Ad;W W ðB AÞWBd;W , then kBd;W k 6 kAd;W k þ kAd;W W ðB AÞW kkBd;W k i.e., kBd;W k 6
kAd;W k : 1 kAd;W W ðB AÞW k
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
415
Finally, since Bd;W Ad;W ¼ Ad;W W ðB AÞW ðAd;W þ Bd;W Ad;W Þ kBd;W Ad;W k 6 kAd;W W ðB AÞW kðkAd;W k þ kBd;W Ad;W kÞ; we get kBd;W k kAd;W k kAd;W W ðB AÞW k 6 : kAd;W k 1 kAd;W W ðB AÞW k
p
p
Proposition 5.2. The condition ðWBÞ ¼ ðWAÞ could be placed with any of the equivalent formulations discussed above. Proposition 5.3. If kAd;W kkW ðB AÞW k < 1 in the proceeding theorem, then (31) becomes kBd;W Ad;W k kAd;W W ðB AÞW k jd;W ðAÞD 6 ; 6 Ad;W 1 kAd;W W ðB AÞW k 1 jd;W ðAÞD ðBAÞW k where jd;W ðAÞ ¼ kAd;W kkWAW k, D ¼ kWkWAW . k
Theorem 5.3. Let B ¼ A þ E 2 C m n , WE ¼ WEðWAÞðWAÞ ðAW ÞEW . Then
D
D
and EW ¼ ðAW Þ
1
Bd;w ¼ ½I þ Ad;W WEW Ad;W :
Proof. We define E11 P 1 EQ ¼ E21 thus, we obtain E11 E¼P E21
E12 ; E22
ð32Þ
E12 Q1 : E22
ð33Þ
So we can apply the matrix E in the above form to the condition WE ¼ WEðWAÞðWAÞD and EW ¼ ðAW ÞD ðAW ÞEW , we can get W11 E11 ¼ W22 E22 ¼ 0 and E21 W11 ¼ E22 W22 ¼ 0 that is E12 ¼ 0, E21 ¼ 0. E11 W11 0 E11 0 P 1 ; E¼P Q1 and EW ¼ P 0 E22 0 0
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M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
Bd;W ¼ ½ðBW ÞD 2 B ¼ ½ðAW þ EW ÞD 2 ðA þ EÞ (
D )2 0 A11 W11 þ E11 W11 1 ¼ P P 0 A22 W22 0 A11 þ E11 P Q1 0 A22 þ E22 ! 1 W111 ðA11 þ E11 Þ11 0 Q1 ¼P 0 0 ¼P ¼P
! 1 1 1 ½I þ ðW11 A11 W11 Þ W11 E11 W11 ðW11 A11 W11 Þ 0 Q1 0 0 ! ! ½I þ ðW11 A11 W11 Þ1 W11 E11 W11 1 0 ðW11 A11 W11 Þ1 0 1 P P Q1 0 I 0 0
¼ ½I þ Ad;W WEW 1 Ad;W :
From Theorem 5.3 now it is not difficult to get the following two corollaries: D
Corollary 5.1. Let B ¼ A þ E 2 C m n and EW ¼ ðAW Þ ðAW ÞEW with k ¼ IndðAÞ. Then Bd;W ¼ Ad;W ½I þ WEWAd;W WAW 1 fI þ WAd;W WAW ½Ad;W ðI þ WEWAd;W Þ1 WAWAd;W WEðI WAWAd;W Þg þ
k1 X
1
Ad;W ðI þ WEWAd;W WAW Þ
i¼0
ðBWAd;W WAW Þ
ðiþ2Þ
i
EW ðI Ad;W WAW Þ½ðI Ad;W WAW Þ B
ðI WAWAd;W Þ:
Proof. The proof is analogous to the proof of Theorem 5.3. Since D EW ¼ ðAW Þ ðAW ÞEW , we can get E21 ¼ 0. E11 W11 E12 E22 E11 E12 P 1 ; E¼P Q1 and EW ¼ P 0 E22 0 0 Bd;W ¼ ½ðBW ÞD 2 B ¼ ½ðAW þ EW ÞD 2 ðA þ EÞ (
D )2 A11 W11 þ E11 W11 E12 E22 A11 þ E11 1 ¼ P P P 0 A22 W22 0
E12 A22 þ E22
Q1
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
W111 ðA11 þ E11 Þ
¼P
1
X
!2
A11 þ E11
E12
417
Q1
A22 þ E22 ! L LðA11 þ E11 Þ E12 þ LW11 X ðA22 þ E22 Þ Q1 ¼P 0 0 ! 1 L 0 þ E Þ E þ W X ðA þ E Þ I ðA 11 11 12 11 22 22 ¼P Q1 Q Q1 ; 0 0 0 0 0
0
0
1
where 1
L ¼ ½W11 ðA11 þ E11 W11 Þ ; and X ¼
l1 X ½ðA11 W11 þ E11 W11 Þ1 2 E12 W22 ðA22 W22 Þi : i¼0
As the similar proof of the Theorem 5.3, we can obtain L 0 1 P Q1 ¼ Ad;W ½I þ WEWAd;W WAW : 0 0 So it is not different to get the result: 1
Bd;W ¼ Ad;W ½I þ WEWAd;W WAW fI þ WAd;W WAW ½Ad;W ðI þ WEWAd;W Þ WAWAd;W WEðI WAWAd;W Þg þ
l1 X
1
Ad;W ðI þ WEWAd;W WAW Þ1
i¼0 ðiþ2Þ
ðBWAd;W WAW Þ ðI WAWAd;W Þ:
i
EW ðI Ad;W WAW Þ½AW ðI Ad;W WAW Þ B
Corollary 5.2. Let B ¼ A þ E 2 C m n , and WE ¼ WEðWAÞðWAÞ IngðAÞ. Then ( Bd;W ¼
D
with k ¼
I þ ½ðI Ad;W WAW ÞEWAd;W WAWAd;W ðI þ WEWAd;W WAW Þ1 þ
l1 X
BðI WAWAd;W Þ½WAðI WAWAd;W Þi WEðI WAWAd;W Þ
i¼0
ðWBWAWAd;W Þ
) ðiþ2Þ
WAWAd;W
1
Ad;W ðI þ WEWAd;W WAW Þ :
418
M. Qin, G. Wang / Appl. Math. Comput. 162 (2005) 403–419
6. Concluding remarks In this paper we study the displacement structure and the perturbation of the W -weighted Drazin inverse of a complex matrix. It is natural to ask if we can extend our results to linear operators in Hilbert space. This will be the future research.
Acknowledgements This project is supported by National Natural Science Foundation of China under grant 19901006 and 10171021.
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