Some Topics on Weighted Generalized Inverse and Kronecker Product of Matrices

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1 Malaysia Soe Joual Topics of Matheatical o Weighted Scieces Geealized (): Ivese 9-22 ad Koece (27) Poduct of Matices Soe Topics o Weighted Geealized Ivese ad Koece Poduct of Matices Zeyad Abdel Aziz Al Zhou & Ade Kilica Depatet of Matheatics Uivesiti Malaysia Teeggau 23 Kuala Teeggau Teeggau. E-ail: zeyad968@yahoo.co ailica@ut.edu.y ABSTRACT I this pape we peset two geeal epesetatios fo the weighted geealized ivese W which exteds ealie esults o the Dazi ivese goup ivese A g ad usual ivese A. The fist oe coces with the atix expessio ivolvig Mooe-Peose ivese A +. The secod oe holds o the Koece poducts of two ad seveal atices. Futheoe soe ecessay ad sufficiet coditios fo Dazi ad weighted Dazi iveses ae give fo the evese ode law ( AB) d = B d ad ( AB) dz = B dr W to hold. Fially we apply ou esult to peset the solutio of esticted sigula atix equatios. Keywods: Koece Poduct Weighted Dazi Iveses Geeal algebaic stuctues Idex. Nilpotet atix. (2 Matheatics Subject Classificatio: 5A69 ; 5A9.) INTRODUCTION AND PRELIMINARY RESULT Oe of the ipotat types of geealized iveses of atices is the weighted Dazi ivese which has seveal ipotat applicatios such as applicatios i sigula diffeetial diffeece equatios Maov chais statistical pobles cotol syste aalysis cuve fittig ueical aalysis ad Koece poduct systes [e.g ]. Hee we use the followig otatios. Let M be the set of all atices ove the coplex ube field ad whe = we wite M istead of M. Fo atix a of A. If that A M let A* be the cojugate taspose of A ad a(a) be the A M is a give atix the the sallest o-egative itege such a(a + ) = a(a ) () is called the idex of A ad is deoted by Id(A)=. Malaysia Joual of Matheatical Scieces 9

2 Zeyad Abdel Aziz Al Zhou & Ade Kilica It is well ow that the Dazi ivese (DI) of A M with Id(A)= is defied to be the uique solutio X M satisfyig the followig thee atix equatios: A XA = A XAX = X AX = XA (2) ad is ofte deoted by X =. Note that the fist equatio i (2) ca be witte as + A X A =. I paticula whe Id (A)= the Dazi ivese of the goup ivese of A ad is ofte deoted by A g but whe Id(A)= ad is a o-sigula atix the = A -. Wag [3] gave that fo A M with Id(A) = A M is called A M A ( Ad ) A = A ( ) ( ) ( ) Ad A Ad = Ad A ( Ad) = ( Ad) A (3) By the uiqueess of the DI we have (A ) d = ( ) (4) Fo oe popeties coceig Dazi iveses see [e.g ]. Clie ad Geville [5] exteded the Dazi ivese of squae atix to ectagula atix ad called it as the weighted Dazi ivese (WDI). The WDI of A M with espect to the atix W M is defied to be the uique solutio X M of the followig thee atix equatios: whee (AW) + XW = (AW) XWAWX = X AWX = XWA (5) = ax {Id(AW) Id (WA)} (6) ad is ofte deoted by X = W. I paticula whe A M ad W = I the w educe to i.e. = I. If A M is o-sigula squae atix ad W = I it is easily see that Id(A) = ad w = = A - satisfies the atix equatios (5). The popeties of WDI ca be foud i [e.g ]. Soe otable popeties ae: If A M with espect to the atix W M ad = ax {Id(AW)Id(WA)} the: i. W = A {(WA) d } 2 = {(AW) d } 2 A (7) Malaysia Joual of Matheatical Scieces

3 Soe Topics o Weighted Geealized Ivese ad Koece Poduct of Matices ii w W= (AW) d WW = (WA) d (8) iii WAWW = WA(WA) d W WAW = (WA) d AW (9) iv Oe closed-fo solutio of w fo a ectagula atix A M l+ 2 2 l li ( AW ) + α I ) ( AW ) A α if l AdW = l l+ 2 2 li AWA ( ) (( WA) + α I) () α if l The Mooe-Peose ivese (MPI) is a geealizatio of the ivese of o-sigula atix to the ivese of a sigula ad ectagula atix. The MPI of a atix A M is defied to be the uique solutio X M of the followig fou Peose equatios: AXA = A XAX =A (AX)* = XA (XA)* = XA () ad is ofte deoted by X = A +. Note that if A M is o-sigula atix the A+ = A -. Regadig vaious basic popeties coceig MPI see [e.g ]. The geeal algebaic stuctues (GAS) of the atices A M W M A + W + ad AdW M with = ax{id(aw)id(wa)} ae (see [e.g. 4922]): A A= L Q A 22 W W = Q L W 22 A = Q L + A (2) W = L Q + W ( WAW) AdW = L Q (3) whee LQA W ae o-sigula atices ad A 22 W 22 A 22 W 22 W 22 A 22 ae ilpotet atices ( A atix A M is called ilpotet if A = fo soe positive itege ). Malaysia Joual of Matheatical Scieces

4 Zeyad Abdel Aziz Al Zhou & Ade Kilica I paticula whe A M with Id(A) = W = I ad L = Q the we have A A= L L A 22 A d A = L (4) whee L ad A ae o-sigula atices ad A 22 is a ilpotet atix. Geville [6] fist studied the evese ode law fo the Dazi ivese of the poduct of two atices A ad B M. He poved that (AB) d = B d holds ude the coditio AB = BA. Tia [2] gave a ecessay ad sufficiet coditio fo the evese ode law (AB) d = B d by usig a a idetity. A siila esult fo evese ode law fo Dazi ivese of geeal ultiple atix poduct was peseted by Wag [4] as follows: Let A ad B M be give with =ax {Id(A) Id(B) Ix(AB)}. The if ad oly if (AB) d =B d (5) 2+ ( ) ( AB) ( AB) 2+ A A a a( A ) a( B ) a(( AB) ) 2+ = + + B B A ( AB) B (6) Fially the Koece poduct of A= a ij M [ a B] M ij p q ij B = b M is give by ad [ l ] p q A B = (7) whee a B M is the ij - th bloc. ij p q Fo ay copatible atices A B C ad D; ad ay eal ube we shall ae fequet use of the followig popeties of the Koece poduct (see [e.g.2722]): i. If AC ad BD ae well defied the. 2 Malaysia Joual of Matheatical Scieces

5 Soe Topics o Weighted Geealized Ivese ad Koece Poduct of Matices ( A B)( C D) = AC BD (8) ii. If A ad B ae squae positive (sei) defiite atices the ( A B ) = A B (9) iii. a ( A B)= a (A)a(B) (2) iv. Vec (AXB T ) = ( B A) Vec (X) (2) whee [ x... x x... x... x x ] T Vec ( X ) =... (22) 2 deotes vectoizatio by colus of abitay atix X M 2 v. If A ad B ae ilpotet aticesthe A B is ilpotet atix vi. If A ad B ae uitay atices the A B is uitay atix. I this pape soe ew atix expessios ivolvig the thee ids of geealized iveses of the Koece poducts atices ae established. I additio by usig the geeal algebaic stuctues of atices (GAS) the ecessay ad sufficiet coditios fo Dazi ad weighted Dazi iveses ae also give fo the evese ode laws (AB) d = B d ad (AB) dz = B dr W to hold. Fially we apply ou esult to peset the solutio of esticted sigula atix equatios (WAW) X (RBR) T = C. MAIN RESULT Obseve that i geeal if A ad B M ae ilpotet atices the AB eed ot be ilpotet. As a exaple let A= B = It is easy to veify that A ad B ae ilpotet atices but AB = is ot ilpotet because ( AB) = AB fo all positive itege. This obsevatio is Malaysia Joual of Matheatical Scieces 3

6 Zeyad Abdel Aziz Al Zhou & Ade Kilica ipotat to give a ilpotet coditio whe we use the GAS of atices ude usual poduct as follows: Theoe : Let A A L = A 22 B B= L B 22 L W W = L W 22 L L (23) R = L Z = Z 22 R Z L R L 22 (24) be the geeal algebaic stuctues espectively of A B W R ad Z M with = ax {Id(AW) Id(WA) Id(BR) Id(RB) Id(ABZ) Id(ZAB)}. The (AB) dz = B dr w (25) if ad oly if A 22 B 22 is a ilpotet atix ad L ( R B R ) ( W A W ) = ( A B Z ) O equivaletly Z (26) R B R W A W = Z B A Z (27) Poof : The GAS of ABW R ad Z i the assuptios assue that A B W R Z ad L ae o-sigula atices ad A 22 B 22 W 22 R 22 ad Z 22 ae ilpotet. The it is well ow that the GAS of W ad B dr ae give by ( W A W ) AdW = L L ( R B R ) = BdR L L (28) Coputatio shows that ( R B R ) ( W A W ) BdR AdW = L L (29) 4 Malaysia Joual of Matheatical Scieces

7 Soe Topics o Weighted Geealized Ivese ad Koece Poduct of Matices ad AB ( Z A B Z ) ( AB) dz = L L = L L A B dz if ad oly if A 22 B 22 is a ilpotet atix. It is clea fo (29) ad (3) that (3) (AB) dz = B dr w if ad oly if A 22 B 22 is a ilpotet atix ad ( R B R ) ( W A W ) = ( Z A B Z ) This copletes the poof of Theoe. If we set W = R = Z =I i Theoe we obtai the sufficiet ad ecessay coditio fo the evese ode of Dazi ivese as follows. Coollay Let A = L L B = L L A 22 B 22 A be the GAS of A ad B M espectively with = ax {Id(A) Id(B) Id(AB)}. The (AB) d = B d if ad oly if A 22 B 22 is a ilpotet atix. Now we ca also apply the GAS i ode to fid a ew epesetatio of WDI as follows: Theoe 2: Let A M ad W M such that A 22 W 22 ad W 22 A 22 ae ilpotet atices of idex i GAS fo. The the WDI of A with espect to the atix W ca be witte as atix expessio ivolvig MPI by 2 { } + B + + AdW = ( AW) ( AW) ( AW) W (3) whee = ax {id(aw) Id(WA)}. + Poof: Due to the GAS of AAWW + ad A dw thee exists o-sigula atices L A ad W ad ilpotet atices A 22 ad W 22 such that A A= L Q A 22 W W = Q L W 22 W = Q Q + W Malaysia Joual of Matheatical Scieces 5

8 Zeyad Abdel Aziz Al Zhou & Ade Kilica Sice A 22 W 22 ad W 22 A 22 ae ilpotet atices of idex the (A 22 W 22 ) = ad it is easy to show that ( AW ) = ( AW ) L L ( ) 2 AW ( AW ) = L Coputatio shows that 2 + {( AW ) [( AW ) ] ( ) } + AW W + = L ( A W ) 2 ( A W ) ( W ) W Q A = = = L L L 2 ( A W ) ( A W ) ( A W ) W ( A W ) W Q ( W A W ) Q Q = A d w This copletes the poof of Theoe 2. If A is a squae atix with Id(A) = ad set W = I i Theoe 2 we obtai the followig coollay which is give by Wag [4]: Coollay 2 Let A M with Id(A)= the 2 + ( ) + d = (32) A A A A Theoe 3 Let A M W M B M ad p q R M be atices with q p { ( ) } ( ) 2 { } = ax Id AW Id( WA) = ax Id BR Id( RB) Also let Z = W R ad = ax { 2 }. The 6 Malaysia Joual of Matheatical Scieces

9 Soe Topics o Weighted Geealized Ivese ad Koece Poduct of Matices { } i. Id ( ) A B Z = (33) A B = A B (34) ii. ( ) dz dw dr Poof: i. By assuptios we have a( AW ) = a( AW ) a( BR) = a( BR) Fo popeties of Koece poducts we have { } { } a ( A B) Z = a ( A B)( W R) = a { AW BR} Siilaly { } = a { AW } a { BR }. a ( A B) Z + = a { } AW + a {BR} +. It is obvious that the sallest o-egative itege such that { } { B Z} a ( A B) Z + = a ( A ). is = ax { 2 }. Hece (33) is tue. ii. Let X = Adw BdR ad Z = W R. Fo popeties of the Koece poduct ad (5) we have (( ) ) ( )( ) + + = (( ) dw ) ( ) ( ) ( dw dr )( ) + + A B Z XZ = A B W R A B W R ( dr ) AW A W BR B R ( ) ( AW ) ( BR) ( AW BR) ( A B)( W R) = = = {( A B) Z} = (35) ( ) = ( dw dr )( )( )( )( dw dr ) XZ A B ZX A B W R A B W R A B ( ) ( ) = A WAWA B RBRB = A B (36) = X. dw dw dr dr dw dr ( A B) ZX = ( A B)( W R)( Adw BdR ) AWA BRB = A WA B RB = dw dr dw dr = ( A B )( W R)( A B) dw d R = XZ ( A B) (37) Malaysia Joual of Matheatical Scieces 7

10 Zeyad Abdel Aziz Al Zhou & Ade Kilica Fo (35)-(37) we ca obtai (34) iediately. If A ad B ae squae atices with Id(A) = ad Id(B) = 2 espectively ad set W = I ad R = I i Theoe 3 we obtai the followig coollay which is give by Wag [3]: Coollay: 3 Let A M ad B M with Id(A) = ad Id(B) = 2 espectively. The A B =ax{ 2 } (38) ad { } Id ( ) ( A B) Ad Bd = (39) d Moe paticulaly if Id(A) = Id(B) = the we have. ( A B) Ag Bg = (4) g Coollay: 4 Let Ai Mi () i () ad Wi M () ()( i 2 i i ) be atices with = ax Id( AW ) Id( W A ) i = The { } i i i i i Id Π Ai Z = ad Π Ai = Π ( Ai) dw i (42) d z (4) whee = ax { } i. if Ai M() i 2... ad Z ad W I ( i 2) i i =Π W. I paticula = we the have i ad Id Π Ai = Π A ( A ) i i i d = d = Π (43) (44) whee = ax {Id (A i ) i = 2...} ii. if Id(A )=Id(A 2 )=...=Id(A )= we the have Π A = Π ( A) i i i g = g (45) 8 Malaysia Joual of Matheatical Scieces

11 Soe Topics o Weighted Geealized Ivese ad Koece Poduct of Matices Poof The poof of (4) is by iductio o. The base case (whe = 2) has bee established i (33) of Theoe 3. I the iductio hypothesis we assue that Id Π Ai Π Wi = I d { Π AW i i } Now = γ = ax { 2... } Π i = Π i Π i Id A Z Id A W { AW i i } = Id Π = Id Π AW i i ( AW ) { γ } { } = ax = ax... = 2 The poof of (42) is also by iductio o. The base case (whe =2) has bee established i (34) of Theoe 3. I the iductio hypothesis we assue that Now Π A e = Π ( A ) i i dw i d Π Wi Π Ai = Π Ai ( A) dw dz d Π e Wi = Π ( Ai) ( ) dw A i dw =Π ( Ai ) dwi The poof of thee special cases i (43)-(45) ae staightfowad. Oe of the ipotat applicatio of Theoe 3 is that the weighted Dazi ivese of Koece poduct aise atually i solvig the so-called esticted sigula atix equatios (RSME) as follows. Theoe 4: Let A M W M B M pq R M ad C M q p q be give costat atices ad X M p be a uow atix to be solved. Also let Malaysia Joual of Matheatical Scieces 9

12 Zeyad Abdel Aziz Al Zhou & Ade Kilica (( ) ) ( ( )) L = R W = Id B A L 2 = Id L B A (46) such that (( ) ) = ( ( )) ( ( )) (( ) ) 2 2 B A L L B A VecC R L B A VecX R B A L (47) The the uique solutio of the followig RSME is give by ( WAW ) X ( RBR) T = C (48) X = A CB (49) T dw dr Poof Usig idetity (2) it is ot difficult to tasfo (48) ito the vecto fo as: ( ( ) ) L B A L VecX = VecC (5) It is easy to veify ude coditios (47) that the uique solutio of (5) is ( ) dl ( dr dw ) = Vec( A T dw CBd R) VecX = B A VecC = B A VecC which is the equied esult. A ipotat paticula case of Theoe 4 is that whe = p = q W = I ad R = I p we obtai the followig coollay: Coollay 5 : Let A M B M p ad C M p be give costat atices ad X M p be a uow atix to be solved. The the uique solutio of the followig RSME T AXB = C : VecC VecX R ( B A) = (5) is give by T X = A CB (52) d d Id( B A) CONCLUSION I this pape we have peseted two geeal epesetatios fo weighted Dazi ivese elated to Mooe-Peose ivese ad Koece poduct of two ad seveal atices. These epesetatios ae viewed as a geealizatio of Wag's esults i [3 Lea. ad 4 Theoe 2.2]. Futheoe soe ecessay ad sufficiet coditios fo Dazi ad weighted Dazi iveses ae give fo the evese ode law (AB) d =B d ad ( AB) = B dz dr Adw to hold. Although the esults ae applied to solve the esticted sigula atix equatios the idea adopted ca be easily exteded to 2 Malaysia Joual of Matheatical Scieces

13 Soe Topics o Weighted Geealized Ivese ad Koece Poduct of Matices solve the coupled esticted sigula atix equatios. It is atual to as if we ca exted ou esults to the Miowsi ivese i Miowsi space. This will be pat of futue eseach. REFERENCES AL ZHOUR Z. ad A. KILICMAN. 26. Matix equalities ad iequalities ivolvig Khati-Rao ad Tacy-Sigh Sus. J. of Ieq. i Pue ad Appl. Math. 7() Aticle 34. AL ZHOUR Z. ad A. KILICMAN. 26. Extesios ad geealizatio iequalities ivolvig the Khati-Rao poduct of seveal positive atices. J. of Ieq. ad Appl. 26 Aticle ID 8878: -2. BEN - ISRAEL A. ad T. N. E. GREVILLE Geealized Iveses: Theoy ad Applicatios. New Yo: Wiley. CAMPBELL S.L. ad C.D. MEYER Geealized Iveses of Liea Tasfoatios. New Yo: Dove Publ. Ic. CLINE R.E. ad T.N.E. GREVILLE. 98. A Dazi ivese fo ectagula atices. Li. Alg. ad its Appl. 29: GREVILLE T.N.E Note o geealized ivese of atix poduct. SIAM Rev. 8: KILICMAN A. ad Z. AL ZHOUR. 25. The geeal coo exact solutios of coupled liea atix ad atix diffeetial equatios. J. Aal. Coput. (): RAKOCEVI C V. ad Y. WEI. 22. A weighted dazi ivese ad applicatios. Li. Alg. ad its Appl. 35: RAKOCEVI C V. ad Y. WEI. 2. The epesetatio theoy fo the Dazi ivese ad its applicatios II. J. Austal. Math. Soc. 7: RAO C.R. ad S.K. MITRA. 97. Geealized Iveses of Matices ad its Applicatios. New Yo: Wiley. STEEB W.-H Matix Calculus ad Koece Poduct with Applicatios ad C ++ Pogas. Sigapoe: Wold Scietific Publishig Co. Pte. Ltd. TIAN Y O the evese ode law (AB) d = B d. J. Math. Res. Expositio 9(2): WANG G Weighted Mooe - Peose Dazi ad Goup Iveses of the Koece Poduct A B ad Soe Applicatios. Li. Alg. ad its Appl. 25: WANG G. 22. The evese ode law fo the Dazi iveses of ultiple atix poducts. Li. Alg. ad its Appl. 348: Malaysia Joual of Matheatical Scieces 2

14 Zeyad Abdel Aziz Al Zhou & Ade Kilica WANG G. ad Y. WEI The iteative ethods fo coputig the geealized ivese A + ad A Nue. Math. J. Chiese Uiv. 6: MN dw WEI Y Achaacteizatio ad epesetatio of the Dazi ivese. SIAM J. Matix Aal. Appl. 7: WEI Y. ad H. WU. 2. The epesetatio ad appoxiatio fo the Dazi ivese. J. Coput. Appl. Math. 26: WEI Y. 22. A chaacteizatio fo the W-Weighted Dazi ivese solutio. Appl. Math. Coput. 25: WEI Y. 23. Itegal epesetatio of the W-Weighted Dazi ivese. Appl. Math. Coput. 44: 3-. WEI Y. C. -W. WOO ad T. LEI. 24. Aote o the petubatio of the W-Weighted Dazi ivese. Appl. Math. Coput. 49: WEI Y. ad D. DJORDJEVI C. 23. O itegal epesetatio of the geealized ivese A (2). Appl. Math. Coput. 42: T S ZHANG F Matix Theoy: Basic Results ad Techiques. New Yo: Spige-Velag. 22 Malaysia Joual of Matheatical Scieces