关于加边矩阵的奇异性及其自反广义逆的结构_英文_

关于加边矩阵的奇异性及其自反广义逆的结构_英文_ On the Structure of Ref lexive G2inverses and Nonsingular ity of A Bordered Ma tr ix GU O Wen2bi n , W E I Mu2sheng ( )Depa rt ment of M at hem at ics , East Chi n a N or m al U ni versi t y , S hanghai 200062 , Chi n a A B M = and give t he Abstract :In t his article we st udy t he singularit y of t he bordered mat rix C 0 DD 1 2- st ruct ure of it s reflxive g - inverses M = by applying t he multiple quotient singular val2 r DD 3 4 ue deco mpositio n QQ - SVD. Key words :reflexive g - inverse ; bordered mat rix ; QQ - SVD. CLC number : O151 . 21 Document code : A 0 Introduction Bo rdered mat rices have been st udied in t he literat ure , and t hese mat rices also play an im2 po rtant role in vario us fields of applied mat hematics , fo r example , quadratic p ro gramming wit h equalit y co nst raint s , so me kinds of t wo - dimensio nal interpolatio n , linear statistical inference , finding generalized inverse of mat rices , etc. U sing t he bo rdering technique , Manjunat ha and given an met ho d of co mp uting mino rs of a reflexive g - inverse of a regu2 K. P. S. Rao 8 have lar mat rix A . In 4 , t he result s o n bo rdered mat rices and used in o rder to co nst ruct invertible 2 g - in“extensio ns”of ceratin t ype of“band”mat rices. Chen and Zho u 3 have st udied t he A B verse and no nsingularit y of a bo rdered mat rix M = w hic satisfies t he rank additivit y 0 C co nditio n A B ( ) ) ( ) ( rank M= rank B + rank C , 0. + rank A , = rank 0 C - In t his paper we will give t he st ruct ure of reflexive g - inverse of a bo rdered mat rix M r A B M = wit ho ut any rest rictio n . C 0 收稿日期 :2001 - 02 () 基金项目 :国家自然科学基金 19871029,上海市重点学科建设项目. () 作者简介 :郭文彬 1973 - ,男 ,博士研究生 ,讲师 ,现在蚌埠坦克学院工作. () 华东师范大学学报 自然科学版2002 年 10 m ×n We use t he following notatio n . L et Cbe t he set of m ×n mat rices wit h co mplex en2 m m ×n H n ×m t ries , Cbe t he set of m - dimensio nal vecto rs. Fo r a mat rix A ?C, A ?Cis t he ( ) ( ) co njugate t ranspo se of A , RA denotes t he range space of A , NA denotes t he null space of ( ) A , rank A denotes t he rank of A . I denotes t he identit y mat rix of o rder k , 0 t he l by k l ×m ) (m mat rix of all zero ent ries , 0 = 0 if no co nf usio n occures , we will drop t he subindex. l l ×l Fo r a given mat rix L , a generalized inverse T of L is a mat rix w hich satisfies so me of t he following fo ur equatio ns : ( ) ( ) 1L TL = L ; 2TL T = T ; ( )1 H H ( ) ( ( ) ( ) ) 3L T = L T ; 4TL = TL . Φ η( ) Then Ldenotes t he set of all mat rices T w hich satisf y i fo r all i L et ?Α{ 1 , 2 , 3 , 4} . η ηηη ?. Any T ?is called an - inverse of L . One usually denotes any { 1} - inverse of L as ( ) - 1L L o r w hich is also called a g - inverse of L . Any { 1 , 2 } - inverse of L is denoted by ( ) 1 , 2- w hich is also called a reflexive g - inverse of L . Fo r a bo rdered mat rix L o r L r m A B ( )2 M = C0 q n , p we denote D D 12n - ( )M 3 = r D D p 34m , q - in t his paper . We will st udy t he st ruct ure of M . r The paper is o rganized as follow s. In sectio n 2 , we give so me equivalent co nditio ns fo r M - M to be a no nsingular mat rix . In sectoin 3 , we give t he exp ressio n of reflexive g - inverse of r a bo rdered mat rix M . Finally in sectio n 4 , we make t he co ncluding remar ks. In t he following we will briefly describe so me necessary result s fo r o ur f urt her discussio n . First o ne is t he Q Q - SVD of { C , A , B } . q ×n m ×n ( ) Lemma 1 . 1 5 , Theo rem 4 . 1 . Fo r given mat rices M ?C, A ?Cand B ? m ×p U wit h app rop riate C, t here exist no nsingular mat rices W , W , V and unitary mat rix 1 2 H ( )sizes , such t hat here we take C = A , A = A and B = A 1 2 3 ( )4 C = W DW and BA = U D W ,= W DV , C 1 2 B 2 A 1 in w hich 1 r 3I 0 0 0 0 0 1 1 rr - r I 10 0 0 0 0 I 0 2 3( )D == 5 C 1 0 0 00 rq - 0 0 0 I 1rr- 2 1 r, n - r 1 10 0 0 0 0 0 q - r 11 2 2 2 2 1 1 1 - r, r- r, r, r- r, n - r- r , rr3 2 3 1 2 3 2 3 1 2 第 4 期 郭文彬 ,等 :关于加边矩阵的奇异性及其自反广义逆的结构 11 1 I 0 0 0 0 0 r3 I 1 0 1 0 0 0 0 1 rr -r 32 2 I 0 0 0 I 0 0 0 0 0 2 2 ( )D== r 6 I r0 0 0 3A 2 0 0 I 0 0 0 2 2 0 0 0 0 m - rrr- 22 3 1 2 21 0 0 r, r - r, r, n - r - r 0 0 0 0 2 1 2 2 1 2 3 m - rr- 3 2 0 0 0 0 0 0 1 1 1 1 2 2 2 2 r, r- r, r- r, r, r- r, n - r- r 3 2 3 1 2 3 2 3 1 2 1 0 0 0 I r3 0 1 0 0 1 0 r- r 2 3 0 0 0 I 2 2 r-( )rD= 7 3B 2 0 0 0 0 3 r 30 I 0 0 3 m - r-r 2 3 0 0 0 0 1 2 3 - r , r, r, p r3 3 3 3 in w hich 1 2 3 1 2 ( ) ()( ) ( ) = r+ r+ r= rank B . 8 r r= rank C, r= r+ r= rank A , 1 2 2 2 3 3 3 3 0 0 0 S 1 0 0 0 0 0 S 0 0 2 Remark. In 5 , t he mat rix has t he fo r m D=in w hich each S fo r i B i 0 0 0 0 0 0 0 S 3 0 0 0 0 i = 1 , 2 , 3 is square diago nal mat rices wit h po sitive diago nal element s w hen r> 0 , o r naught 3 i w hen r= 0 and U , V are unitary mat rices. To simplif y notatio n and discussio n in t he paper , 3 we describe L emma 1 . 1 w hich is different f ro m o ne in 5 . We re place unitary mat rix V in () ( y o ne appearing in 7. )5 b y diag S , S , S ,V and replace Din 5 bI 1 2 3 B p - r 3 Fro m L emma 1 . 1 we have q ×n m ×n m ×p Corollary 1 . 1 Suppo se t hat mat rices C ?C, A ?Cand B ?C. L et t he Q Q () () - SVD of t he t riple { C , A , B } be as in 4- 7. Then t he following statement s hold : 1 2 () ( ) i. rank A = r= r+ r, 2 2 2 1 2 3 ( ) rank B = r= r+ r+ r, 3 3 3 3 A 2 3 ( ) ( ) = r+ r, rank A , B = r + r, rank C= r, rank 1 1 2 2 3 C 2 1 3 ( ( )) rank M = r+ r+ r+ r, 9 2 3 3 1 A () ( ) ( ) ( ) ( ) ii. rank M = rank A , B + rank C= rank + rank B , C 1 1 2 ( )Ζ r= r, r= 0 . 10 2 3 3 ()() Proof . Fro m L emma 1 . 1 , o ne can o btain i easily. Taking iinto acco unt , () iiholds immediately. () 华东师范大学学报 自然科学版2002 年 12 q ×n m ×n m ×p Corolary 1 . 2 Suppo se t hat mat rices C ?C, A ?Cand B ?C. L et t he Q Q () () - SVD of t he t riple { C , A , B } be as in 4- 7. Then t he following statement s are equiva2 lent : HH() ( ) ( ) ( ) ( ) i. NA ?NC= { 0} and NA ?NB = { 0} ;()11 2 3 () ii. n = r+ r, m = r + r. 1 2 2 3 Proof . Fro m L emma 1 . 1 , it is o bvio us t hat - 1 ( ) ( ) ( ) NA = NW DW = W ND , 2 A 1 1 A - 1 ( ) ( ) ( ) NC= NU D W = W ND . C 1 1 C On t he ot her hand , we can easily o btain T T T T ( ) ( ) ND = { x | x = 0 0 0 x } , x x C4 5 6 in w hich 2 2 2 2 rr- rn - r- r3 2 3 1 2 x ?C, x ?C, x ?C; 4 5 6 and T T T ( ) ( ) ND= { y| y = 0 0 y 0 0 y } , A 3 6 in w hich 1 2 r- rn - r- r1 2 1 2 y ?C, y ?C. 3 6 Then T T ( ) ( ) ( ) ND?ND = { x | x = 0 0 0 0 0 x } ,A C6 2 n - r- r1 2 in w hich x ?C, so 6 2 ( ) ( ) ( ) ( ) NA ?NC= ND?ND = { 0} Ζ n = r+ r.2 A C1 Wit h t he same reaso n , we have HH3 ( ) ( ) NA ?NB = { 0} Ζ m = r+ r.3 2 To simplif y co mp utatio ns and discussio n , we rew rite M in ter ms of t he exp ressio n of M in L emma 1 . 1 as follow s. q ×n m ×n m ×p Corollary 1 . 3 Suppo se t hat mat rices C ?C, A ?Cand B ?C. L et t he Q Q () () - SVD of t he t riple { C , A , B } be as in 4- 7. Then - 1 - 1 PPQ Q W W 0 0 I 0 1112111221 ( ) M = 120 0 PPQ Q 0 U 0 V 21 22 21 22 in w hich 第 4 期 郭文彬 ,等 :关于加边矩阵的奇异性及其自反广义逆的结构 13 1 0 0 0 0 0 0 I 0 0 0 r3 0 0 0 0 0 0 0 I 0 0 a 1 0 0 0 0 0 0 0 0 I 0 r - r1 2 2 0 0 0 0 0 0 I 0 0 0 r3 2 2 0 0 0 0 0 0 0 I 0 0 - r rPP11 12 2 3 P = = 3 PP21220 0 0 0 I 0 0 0 0 0 r3 1 1 I 0 0 0 0 0I 0 0 0 - rr3 3 0 0 0 0 0 0 0 0 0 I b 0 I 0 0 0 0 0 - I 0 0 a a 0 0 0 0 0 0 0 0 0 I q - r1 1 0 0 0 0 I 0 0 0 0 0 r3 0 0 0 0 0 I 0 0 0 0 a 1 0 0 I 0 0 0 0 0 0 0 r - r1 2 2 2 0 0 0 I 0 00 - I 0 0 rr3 3 2 2 0 0 0 0 0 0 0 0 I 0r - r Q Q 2 2 311 1 Q = = Q Q 0 0 0 0 0 0 0 0 I0 2122 c 1 0 0 0 0 0 0 I 0 0 0 r3 2 0 0 0 0 0 0 0 I 0 0 r3 30 0 0 0 0 0 0 0 0 I r3 0 0 0 0 0 0 0 0 0 I q - r3 1 1 3 2 r - wit h a = r- r, b = m - r - r, c = n - r. 2 3 2 3 1 2 I 0 I X l ×ll ×l- Lemma 1 . 2 Suppo se t hat N = , t hen = w here X , Y are N , r 0 0 Y Y X arbit rary wit h denoted dimensio ns. 1 Conditions f or M to be nonsingular Suppo se t hat m + q = n + p , i . e . M is a square mat rix , t hen it is impo rtant w het her M is no nsingular . In t he following , we give so me equivalent co nditio ns fo r M to be no nsingular . A B m ×n m ×p q ×n . L et Theorem 2 . 1 Suppo se t hat A ?C, B ?C, C ?Cand M = 0 C t he Q Q - SVD of t he t riple { C , A , B } be given in L emma 1 . 1 . Then t he following statement s are equivalent : () i. M is no nsingular ; 2 1 3 () ( ) ii. rank M = r+ r+ r+ r= m + q = n + p ; 1 2 3 3 1 1 2 3 2 () ()iii. r= r, r= 0 , r = q , r = p , m = r + r, n = r + r; 13 2 3 3 1 3 2 3 1 2 () 华东师范大学学报 自然科学版2002 年 14 A () ( ) ( ) ( ) ) ( iv. rank M = rank A , B + rank C= rank + rank B , C B is f ull column rank , C is f ull row rank , HH( ) ( ) ( ) ( ) NA ?NB = { 0} and NA ?NC= { 0} . ( ( ( ( ) ) ) )Proof . Fro m Co rollary 1 . 1 and Co rollary 1 . 2 , we know t hat i Ζ i i and i i i Ζ i v ( ) ( ) i Ζ i i i . immediately , so we o nly p rove t hat Fro m Co rollary 1 . 3 , we have - 1 - 1 PPQ Q W W 0 0 0 1112I 111221 M = , PP0 Q Q 0 0 U 0 V 21 22 21 22 so if M is no nsingular , t hen ( )r M 0 0 0 I 3 rm - r- 3 2 0 0 0 0 0 I = 1 1 0 0 0 0 0 0 r- r2 3 0 0 0 0 q - r 12 2 ( ) r M , r, n - r - r, p - r 3 1 2 3 must be no nsingular , so 3 1 1 m = r+ r, r= r, q = r 2 3 2 3 1 2 2 r= 0 , n = r+ r, p = r. 3 1 2 3 On t he co nt rary , if 3 1 1 m = r+ r, r= r, q = r 2 3 2 3 1 2 2 r= 0 , n = r+ r, p = r, 3 1 2 3 I 0 ( )r M I , so M is nonsingular. We t hen complete t he proof of t he t heorem. t hen = r ( M ) 0 0 2 Ref lexive G - inverses of M In t his sectio n we will derive a general fo r mula of any reflexive g - inverse of t he bo rdered mat rix M . In ter ms of L emmas 1 . 1 , 1 . 2 and Co rollary 1 . 3 , we first state A B m ×n m ×p q ×n Theorem 3 . 1 Suppo se t hat A ?C, B ?C, C ?Cand M = . L et 0 C - t he Q Q - SVD of t he t riple { C , A , B } be given in L emma 1 . 1 . Then any g - inverse M of r M has t he following fo r m. - 1 - 1 D D W W 0 0 12E F 1 2 - ( )M = D = = 14 r - 1 H G H D D 340 0 V U n ×m n ×q p ×m p ×q in w hich E ?C, F ?C, G ?C, H ?C, P I X 11 ( )E = Q Q 1112 Y Y X P21 第 4 期 郭文彬 ,等 :关于加边矩阵的奇异性及其自反广义逆的结构 15 1 r 30 X 0 0 0 X 1211 1 1 r- r 2 0 X 0 0 0 X 322 21 1 0 0 0 0 rX r- 2 X 31 1 32 ( )= 15 2 Y EY Y Er- I - - - Y 17 42 15 16 46 14 3 2 2 0 0 X 0 X I 51 52 r- r 2 3 2 Y Y EEY Y 27 26 66 622425rn - r- 2 1 1 1 1 2 2 2 3 3 - r, r, r- r, r, m - r- r , rr3 2 3 3 2 3 3 2 3 1 r 3I - X X 0 1213 1 1 r- r 2 I - X 0 X 30 22 23 1 I rP0 X r- - 2 12 X 33 I X 1 32 )( )( Q = 16 F = Q 12 112 Y - Y Y X Y FY Fr- P17 13 44 22 11423 2 2 0 0 X - X 53 52 r- r 2 3 2 Y - Y Y FF 21 23 276264 rn - r- 2 1 1 1 1 1 - r, r- r, q - r , rr3 2 3 1 2 11 r X 0 X 30 0 I 7271 2 PY GY Y Y Gr 17 22 14 15 16 I 1126 3X ( )= ( )G = Q Q 17 2122 3 I Y X 0 X Y PX 0 0 61r21 62 3 GY Y Y Y G p - 42343537 36 46 r 32 2 2 3 3 1 1 1 - r, r, r- r, r, m - r- r , rr3 2 3 3 2 3 3 2 31 r - I X 0 X 3- 7273 2 Y - PY HY Hr 11 22 13 24 I 3 1217X ( )( )H = Q Q = 18 2122 3 Y X 0 0 X - Y PX 63 r22 62 3 Y HY - Y H 374231 33 44 p - r 31 1 1 1 - r, r- r, q - r , rr3 2 3 1 2 1 ( ) ( ) ( ) w here Pand Q i , j = 1 , 2are denoted by Co rollary 1 . 3 , X = X andY = Y kl 3 ×7 i j ij ij 7 ×3 are arbit rary mat rices wit h X , Y having app rop riate dimensio ns , and E, F,G,Hare ij kl i j ij ij ij denoted as follow s : 7 7 - - X X EY X ,EY X ,= = 42 41 421 k k2 461 k k1 6 6 k = 1 k = 1 ( ) 197 7 EX ,EX Y Y = = 622 k k2 662 k k16 6 k = 1 k = 1 7 7 F= Y X -F= Y X , X - Y , X - 421 k k2 42 12 4443 1 k k3 6 6 k = 1 k = 1 ( )20 7 7 FY X ,EY X = Y - = 22 622 k k2 642 k k36 6 k = 1 k = 1 7 7 ( )Y X ,Y X , fo ri = 2 , 4 ; G= G= 21 i - 1 , k k2 i - 1 , k k1 i2i66 6 k = 1 k = 1 7 7 ( )22 HY -Y X , HY X , fo ri = 2 , 4 . = = i2 i - 1 , 2 i - 1 , k k2 i4i - 1 , k k3 6 6 k = 1 k = 1 () 华东师范大学学报 自然科学版2002 年 16 Proof . Fro m Co rollary 1 . 3 , we know - 1 - 1 PPQ Q 0 0 W W I 12 0 11121121 ( )23 M = 0 0 PPQ Q 0 U 0 V 21 22 21 22 () Then f ro m L emma 1 . 2 and 1, we o btain D D 12- M = r D D 34 - 1 - 1 Q Q PP 0 0 W I X W 11121112( )1 r M 2 = - 1 H Q Y Y X Q PP21 V U 22 21 22 0 0 - 1 - 1 0 0 W W E F 1 2 = - 1 H G H V 0 0 U in w hich P I X 11 ( )( )25 Q Q E = 11 12 Y X PY 21 P12 I X ( )( )F = 26 Q Q 11 12 PY Y X 22 P11 I X ( )( )G = 27 Q Q 21 22 PY X Y 21 PI 12 X ( )( )28 G = Q Q 21 22 Y PY X 22 ( ) ( ) ( ) w here Pand Q i , j = 1 , 2are denoted by Co rollary 1 . 3 , and X and Y are ar2 ij ij ij 7 ×3 kl 3 ×7 () () bit rarymat rices denoted by Theo rem 3 . 1 . Expanding 25- 28, we t hen o btain Theo rem 3 . 1 . - - ) ) M ( ( wit h t he Due to r M = r M , so we can give t he range space and null space of r r - exp ressio n of M . r A B m ×n m ×p q ×n . L et A ?C, B ?C, C ?Cand M = Corollary 3 . 1 Suppo se t hat 0 C DD 12- = be any reflexive g - inverse of M w hich is partitio ned co nfo r ming wit h t hat M r DD 34 of M . L et t he Q Q - SVD of t he t riple { C , A , B } be given in L emma 1 . 1 . Then - 1 W I 1 0 - ( ) M = Q R Rr - 1 0 V Y ( )29 W 20 - 1 - ( ) ( ) NI X . P NM = r 0 U Q PPQ 12 111211 w here P = and Q = are denoted by L emma 1 . 3 , X , Y are arbit rary PPQ Q 21 22 21 22 mat rices denoted by Theo rem 3 . 1 . 3 Concl uding Remarks Because t he reflexive g - inverse T of amat rix L satisf y t he equatio n TL T = T w hich is 第 4 期 郭文彬 ,等 :关于加边矩阵的奇异性及其自反广义逆的结构 17 no nlinear equatio n , so t he st ruct ures of reflexive g - inverses of a bo rdered mat rix by applying Q Q - SVD . Fro m t he exp ressio n of t he st ruct ure of reflexive g - inverses ,we will st udy so me p roperties of reflexive g - inverses of a bo rdered marix near t he f ut ure . Ref erences 1 ] A Ben - Israel , T N E Greville . 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Equalities andinequalities fo r ranks of mat ricesJ . Linear and Multilinear Al gebra , 1974 , 2 : 9 ] 269～292 . S K Mit ra . Properties of t he f undamental bo rderedmat rix used in linear estimatio n M . New Yo r k : No rt h - Holland , 10 ( ) In Statistics and Pro babilit y , Essays in Ho no r of C R Rao G Kallianp ur et al . , Eds. , 1982 , 505～509 . 11 C R Rao , S K Mit ra . Generalized Inverses of Mat rices and It s Applicatio ns M . New Yo r k : Jo hn Wile y & So ns , 1971 . 关于加边矩阵的奇异性及其自反广义逆的结构 郭文彬 , 魏木生 () 华东师范大学 数学系 ,上海 200062 摘要 :作者运用多个矩阵的商型奇异值分解 QQ - SVD ,研究加边矩阵 A B M = C 0 的奇异性 ,并给出它的自反广义逆 DD 12- M = r DD 34 的结构 。 关键词 :自反广义逆 ; 加边矩阵 ; QQ - SVD 中图分类号 :O151 . 21 文献标识码 :A