Characterization of the W-weighted Drazin inverse over the quaternion skew field with applications

Transcription

1 Electronic Journal of Linear lgebra Volume 26 Volume rticle Characterization of the W-weighted Drazin inverse over the quaternion skew field with applications Guang Jing Song sgjshu@yahoocomcn Follow this and additional works at: Recommended Citation Song Guang Jing 2013 "Characterization of the W-weighted Drazin inverse over the quaternion skew field with applications" Electronic Journal of Linear lgebra Volume 26 DOI: This rticle is brought to you for free and open access by Wyoming Scholars Repository It has been accepted for inclusion in Electronic Journal of Linear lgebra by an authorized editor of Wyoming Scholars Repository For more information please contact scholcom@uwyoedu

2 Volume 26 pp 1-14 January CHRCTERIZTION OF THE W-WEIGHTED DRZIN INVERSE OVER THE QUTERNION SKEW FIELD WITH PPLICTIONS GUNG JING SONG bstract characterization of the W-weighted Drazin inverse as well as some determinantal representations of the W-weighted Drazin inverse over the quaternion skew filed are derived Moreover a Cramer s rule for finding the unique solution of a class of restricted quaternion matrix equations is obtained Key words Quaternion matrix W-weighted Drazin inverse Cramer s rule MS subject classifications Introduction Throughout we denote the real number field by R the set of all m n matrices over the quaternion algebra H = {a 0 +a 1 i+a 2 j +a 3 k i 2 = j 2 = k 2 = ijk = 1 a 0 a 1 a 2 a 3 R} by H m n and the identity matrix with the appropriate size by I For H m n the symbol stands for the conjugate transpose of For H n n with k = Ind the smallest positive number such that r k+1 = r k the Drazin inverse of denoted by D is defined to be the unique matrix X that satisfying the equations 1 XX = X 2 k X = k 3 X = X If k = 1 then X is called the group inverse of and is denoted by X = g The Drazin inverse is very useful in various applications see eg [1 [4; applications in singular differential and difference equations Markov chains and iterative methods In 1980 Cline and Greville [5 extended the Drazin inverse of square matrix to rectangular matrix which can be generalized to the quaternion algebra as follows For H m n and W H n m the W-weighted Drazin inverse of denoted by dw is the unique solution to equations W k+1 XW = W k XWWX = X WX = XW for some k > 0 Received by the editors on June ccepted for publication on November Handling Editor: Bryan L Shader School of Mathematics and Information Sciences Weifang University Weifang PR China sgjshu@yahoocomcn Supported by grants from Natural Science Foundation of Shan Dong Province no ZR2010L107 and the PhD Foundation of Weifang University no 2011BS12 1

3 Volume 26 pp 1-14 January Guang Jing Song Researchon the generalizedinverse dw has been actively ongoingfor more than 30 years see eg [6 [13 In 2002 Wei [11 presented a characterization for the W- weighted Drazin inverse as well as a Cramer s rule for the W-weighted Drazin inverse solution of a singular linear equation; in 2004 Wei Woo and Lei [13 established the perturbation bound for the W-weighted Drazin inverse of rectangular matrices under a weaker condition; Wang and Sun [6 derived a Cramer s rule for finding the unique solution of a class of restricted matrix equations In this paper we mainly consider the characterization of the W-weighted Drazin inverse over the quaternion skew field The paper is organized as follows In Section 2 we start with some basic concepts about the row and column determinants of a quaternion matrix In Section 3 we give a characterization of the W-weighted Drazin In Section 4 we present some determinantal representations of the W-weighted Drazin inverse over the quaternion skew field s applications in Section 5 we prove a Cramer s rule for finding the unique solution of a class of restricted matrix equations inverse by the generalized inverse 2 TS 2 Preliminaries The definition of the determinant of a square matrix plays a key role in representing the solution of a system of linear equations Unlike multiplication of real or complex numbers multiplication of quaternions is not commutative Many authors had tried to give the definitions of the determinants of a quaternion matrix see eg [15 [19 Unfortunately using their definitions it is impossible for us to give a determinant representation of an inverse matrix In 1991 LX Chen [20 [21 offered a new definition and obtained a determinant representation of an inverse matrix over the quaternion skew field However this determinant cannot be expanded by cofactors along an arbitrary row or column with the exception of the n-th row Therefore he has also not obtained the classical adjoint matrix or its analogue Recently Kyrchei [22 [24 defined the row and column determinants of a square matrix over the quaternion skew field Suppose S n is the symmetric group on the set I n = {1n} Definition 21 Definition 24 in [22 The i-th row determinant of = a ij H n n is defined by rdet = 1 n r a iik1 a ik1 i k1+1 a ik1 +l i 1 i a ikr i kr+1 a ikr+lr i kr σ S n for all i = 1n The elements of the permutation σ are indices of each monomial The left-ordered cycle notation of the permutation σ is written as follows: σ = ii k1 i k1+1 i k1+l 1 i k2 i k2+1 i k2+l 2 i kr i kr+1 i kr+l r

4 Volume 26 pp 1-14 January Characterization of the W-Weighted Drazin Inverse Over the Quaternion Skew Field 3 Index i opens the first cycle from the left and other cycles satisfy the following conditions i k2 < i k3 < < i kr and i kt < i kt+s for all t = 2r and s = 1l t Definition 22 Definition 25 in [22 The j-th column determinant of = a ij H n n is defined by cdet = 1 n r a jkr j kr+lr a jkr+1 j kr a jjk1 +l j 1 a jk1 +1j k1 a jk1 j τ S n for all j = 1n The elements of the permutation τ are indices of each monomial The right-ordered cycle notation of the permutation τ is written as follows: τ = j kr+l r j kr+1j kr j k2+l 2 j k2+1j k2 j k1+l 1 j k1+1j k1 j Index j opens the first cycle from the right and other cycles satisfy the following conditions j k2 < j k3 < < j kr and j kt < j kt+s for all t = 2r and s = 1l t Suppose that j b denotes the matrix obtained from by replacing its j-th columnwiththecolumnband i bdenotesthematrixobtainedformbyreplacing its i-th row with the row b Theorem 23 Theorem31in [22 If a matrix = a ij H n n is Hermitian then rdet 1 = = rdet n = cdet 1 = = cdet n R Thus we define the determinant of a Hermitian matrix by putting det = rdet i = cdet i for all i = 1n The following theorem about determinantal representation of an inverse matrix of Hermitian follows immediately: Theorem 24 Theorem 51 in [22 If a Hermitian matrix = a ij H n n is such that det 0 then there exist a unique right inverse matrix R 1 and a unique left inverse matrix L 1 and R 1 = L 1 =: 1 They possess the following determinantal representations: R 11 R 21 R n1 L 11 L 21 L n1 R 1 = 1 R 12 R 22 R n2 det L 1 = 1 L 12 L 22 L n2 det R 1n R 2n R nn L 1n L 2n L nn Here R ij L ij are the right and left ij-th cofactors of respectively for all ij = 1n Definition 25 Definition 72 in [22 For H n n the determinant of its corresponding Hermitian matrix is called its double determinant ie ddet := det = det

5 Volume 26 pp 1-14 January Guang Jing Song Theorem 26 Theorem 81 in [22 necessary and sufficient condition of invertibility of H n n is ddet 0 In this case there exist 1 = L 1 = R 1 where L 11 L 21 L n1 L 1 = 1 1 L 12 L 22 L n2 = ddet L 1n L 2n L nn R 11 R 21 R n1 R 1 = 1 1 R 12 R 22 R n2 = ddet R 1n R 2n R nn and L ij = cdet j j a i R ij = rdet i i a j for all ij = 1n 3 Characterization of the W-weighted Drazin inverse over quaternion skew field We begin this section with the follow definitions Definition 31 [14 For an arbitrary matrix H m n we denote by R r = {y H m : y = xx H n } the column right space of N r = {x H n : x = 0} the right null space of R l = {y H n : y = xx H m } the column left space of N l = {x H m : x = 0} the left null space of Definition 32 [14 1 n out inverseof a matrix H m n with prescribed right range space T 1 and right null space S 1 is a solution of the restricted matrix equation XX = X R r X = T 1 N r X = S 1 and is denoted by X = 2 r T1 S 1 2 n out inverse of a matrix H m n with prescribed left range space T 2 and left null space S 2 is a solution of the restricted matrix equation XX = X R l X = T 2 N l X = S 2 and is denoted by X = 2 l T2 S 2 3 n out inverse of a matrix H m n with prescribed right range space T 1 right

6 Volume 26 pp 1-14 January Characterization of the W-Weighted Drazin Inverse Over the Quaternion Skew Field 5 null space S 1 left range space T 2 and left null space S 2 is a solution of the restricted matrix equation XX = X R r X = T 1 N r X = S 1 R l X = T 2 N l X = S 2 and is denoted by X = 2 T 1 T 2 S 1 S 2 The follow lemmas due to Cline and Greville [5 can be generalized to H Lemma 33 Let H m n W H n m and IndW = k Then W D = W D 2 W and IndW k +1 Lemma 34 Let H m n W H n m and IndW = k Then dw = W D 2 = W D 2 We can now prove the following result Theorem 35 Let H m n W H n m k 1 = IndW k 2 = IndW and k = maxk 1 k 2 Then dw = WW 2 R rw k R lw k N rw k N lw k Proof We first need to show dw WW dw = dw pplying Lemmas 33 and 34 to this equation we have Next since dw WW dw = W D 2 WW W D 2 R r dw = R r W D 2 = W D W D = W D 2 WW D = W D 2 = dw R r W D = R r W k

7 Volume 26 pp 1-14 January Guang Jing Song and R r W D = R r W D 2 W R r W D = R r dw it follows that R r dw = R r W k Similarly we have R l dw = R l W k N r dw = N r W k N l dw = N l W k Corollary 36 Suppose that H n n and k = Ind Then for the Drazin inverse D we have D = 2 R r k R l k N r k N l k 4 Determinantal representations of the W-weighted Drazin inverse over the quaternion skew field In this section we will establish determinantal representations of the W-weighted Drazin inverse over the quaternion skew field Theorem 41 Let H m n W H n m with IndW = k 1 IndW = k 2 k = max{k 1 k 2 } and r W k = s Suppose that B H n n s n s and C H m m s m s are of full column ranks and R r B = N r W k N r C = R r W k R l C = N l W k N l B = R l W k Denote M = [ WW B C 0 Then the W-weighted Drazin inverse dw = a ij H m n has the following determinantal representations: 41 or 42 a ij = a ij = m+n s L ki m kj detm i = 1m j = 1n M k=1 m+n s m ik R jk detmm i = 1m j = 1n k=1

8 Volume 26 pp 1-14 January Characterization of the W-Weighted Drazin Inverse Over the Quaternion Skew Field 7 where L ij are the left ij-th cofactors of M M and R ij are right ij-th cofactors of MM respectively for all ij = 1m+n s Proof Set [ X = dw I dw WWC B I WW dw B WW dw WW WWC Then [ MX = Since WW dw +BB I WW dw I BB I WW dw WWC C dw CI dw WWC N r C = R r W k = R r dw = R r [ W D 2 it follows [ C dw = C W D 2 = 0 Moreover since BB = P RrB = P NrW k and BB I WW dw = P NrW k P N rw k WWR rw k = P NrW k WWR rw k we have WW dw +BB I WW dw = P WWRrW k N rw k +P N rw k WWR rw k = I WW I dw WWC BB I WW dw WWC = I WW dw WWC I WW dw WWC = 0 and CI dw WWC = CC = I ItfollowsthatMX = I Hence M isinvertibleaswellasthehermitianmatrixm M Multiplying M M 1 on the right by M we obtain the left inverse LM 1 =

9 Volume 26 pp 1-14 January Guang Jing Song M M 1 M By Theorem 22 we have LM 1 = LM M 1 M L 11 L 21 L m+n s1 1 L 12 L 22 L m+n s2 = detm M L 1m+n s L 2m+n s L m+n sm+n s M 1 = detm M k L k1m k1 k L k2m k1 k L km+n sm k1 k L k1m k2 k L k1m km+n s k L k2m k2 k L k2m km+n s k L km+n sm k2 k L km+n sm km+n s Then dw = a ij has the following determinantal representations: a ij = m+n r L ki m kj detm i = 1m j = 1n M k=1 where L ij are the left ij-th cofactors of M M proving 41 Next we prove formula 42 There exist the inverse MM 1 of the Hermitian matrix MM Multiplying it on the left by M we obtain the left inverse RM 1 = M MM 1 By Theorem 24 we have RM 1 = M RMM 1 = M 1 detmm R 11 R 21 R m+n s1 R 12 R 22 R m+n s2 R 2m+n s R m+n sm+n s R 1m+n s = 1 detmm k m 1k R 1k k m 1k R 2k k m 1k R m+n sk Then 42 follows immediately k m 2k R 1k k m m+n sk R 1k k m 2k R 2k k m m+n sk R 2k k m 2k R m+n sk k m m+n sk R m+n sk Corollary 42 Let H m m with r = s and Ind = k Suppose B H m m s m s and C H m m s m s are of full column rank such that R r B = N r k N r C = R r k R l C = N l k N l B = R l k

10 Volume 26 pp 1-14 January Characterization of the W-Weighted Drazin Inverse Over the Quaternion Skew Field 9 Denote M = [ B C 0 Then the generalized inverse D = a ij H m m has the following determinantal representations: or a ij = a ij = 2m s L ki m kj detm ij = 1m; M k=1 2m s k=1 m ik R jk detmm ij = 1m where L ij are the left ij-th cofactors of M M and R ij are right ij-th cofactors of MM respectively for all ij = 12m s 5 Cramer s rule for the W-weighted Drazin inverse solution In 1970 Robinson [25 gave an elegant proof of Cramer s rule over the complex number field: rewriting x = b as Ii x = i b where I is an identity matrix of order n and taking determinants det detii x = deti b Since detii x = x i i = 1n it follows that x i = deti b i = 1n det which is Cramer s rule This trick of Robinson has been used to derive a series of Cramer s rule for matrix equations see eg [26 [29 fter that Cramer s rules for representations of generalized inverses and solutions of some restricted equations were studied by many authors eg Cai and Chen [30 gave a determinantal representation for the generalized inverse 2 TS and derived some applications; in Chapter 3 of [31 Wang Wei and Qiao surveyed the recent results on the Cramer s rules Recently Kyrchei [23 gave a Cramer s rule for the solution of nonsingular quaternion matrix equation 51 XB = C

11 Volume 26 pp 1-14 January Guang Jing Song as follows Lemma 51 Suppose that B C H n n are given and X H n n is unknown If det 0 and detbb 0 then 51 has a unique solution and the solution is x ij = rdet jbb j c i det detbb or where x ij = cdet j i c Ḅ j det detbb c i := [ cdet i i d 1 cdet i i d n is the row vector and c Ḅ j := [ rdet j BB j d 1 rdet j BB j d n T is the column vector and d i and d j are the i-th row vector and j-th column vector of CB respectively for all ij = 1n In this section we mainly consider the Cramer s rule for finding the unique solution of the following restricted matrix equation 52 W 1 W 1 XW 2 BW 2 = C R r X R r W 1 k1 N r X N r W 2 B k2 R l X R l BW 2 k2 N l X N r W k1 Theorem 52 1 Suppose that H m n W 1 H n m with IndW 1 = k 11 IndW 1 = k 12 k 1 = max{k 11 k 12 }r W 1 k1 = s 1 B H p q W 2 H q p with IndW 2 B = k 21 IndBW 2 = k 22 k 2 = max{k 11 k 12 }r BW 2 k2 = s 2 Suppose L 1 H n n s1 n s 1 H p p s2 p s 2 are of full column rank such that and M 1 H m m s1 m s 1 L 2 H q q s2 q s 2 R r L 1 = N r W 1 k1 N r M 1 = R r W 1 k1 R l M 1 = N l W 1 k1 N l L 1 = R r W 1 k1 R r L 2 = N r W 2 B k2 N r M 2 = R r BW 2 k2 R l M 2 = N l BW 2 k2 N l L 2 = R l BW 2 k2 and M 2

12 Volume 26 pp 1-14 January Denote Characterization of the W-Weighted Drazin Inverse Over the Quaternion Skew Field 11 1 = [ W1 W 1 L 1 M 1 0 [ W2 BW B 1 = 2 L 2 M 2 0 [ C 0 C 1 = 0 0 If C R r W 1 k1 BW 2 k2 and C R l W 1 k1 W 2 B k2 then the restricted matrix equation 52 has a unique solution 53 X = dw CB dw and has the following determinantal representations: rdet j B 1 B1 j c 1 54 i x ij = det 1 1detB 1 B1 i = 1n j = 1p or cdet j 1 1 i c B1 55 j x ij = det 1 1detB 1 B1 i = 1n j = 1p where c 1 i := [ cdet i 1 1 i d 1 cdet i 1 1 i d n is the row vector and [ c B1 j := rdet j B 1 B1 j d 1 rdet j B 1 B1 j d n is the column vector and d i d j are the i-th row vector and jth column vector of 1 C 1B 1 respectively for all i = 1m+n s 1 j = 1p+q s 2 Proof The proof contains two parts We first establish that the unique solution of equation 52 can be expressed as 53 From the definition of the left range and null space of a pair of matrices we have C = W 1 k1 Y BW 2 k2 for some matrix Y H n p It follows that R r C R r W 1 k1 R l C R l BW 2 k2 Then we have W 1 W 1 dw CB dw W 2 BW 2 = P RrW 1 k 1 NrW 1 k 1 CP RlBW 2 k 2 NlBW 2 k 2 = C saying that 53 is a solution of 52 and also satisfies the restricted conditions R r dw CB dw R r dw = R r W 1 k1 N r dw CB dw N r B dw = N r W 2 B k2 R l dw CB dw R l B dw = R l BW 2 k2 N l dw CB dw N l dw = N r W k1 T

13 Volume 26 pp 1-14 January Guang Jing Song For the uniqueness if X 0 is a solution of 52 then and it follows that R r X 0 R r W 1 k1 N r X 0 N r W 2 B k2 R l X 0 R l BW 2 k2 N l X 0 N r W k1 dw CB dw = dw W 1 W 1 X 0 W 2 BW 2 B dw = X 0 Next we establish a Cramer s rule for solving equation 52 Since X is the solution of 52 then we have R l X R l BW 2 k2 = N l M 2 R r X R r W k1 = N r M 1 it follows that XM 2 = 0 M 1 X = 0 and 56 [ W1 W 1 L 1 M 1 0 [ X [ W2 BW 2 L 2 M 2 0 = [ C From the proof of Theorem 41 we have the coefficient matrices of 56 are nonsingular and [ [ [ 1 [ [ 1 X 0 dw CB dw 0 W 1W 1 L 1 C 0 W 2W 2 L 2 = = M M 2 0 Then by Lemma 51 we can get equations Corollary 53 1 Suppose that H n n Ind = k 1 r k1 = r 1 < n B H p p IndB = k 2 rb k2 = r 2 < p and U 1 H n n r1 n r 1 V1 H n n r1 n r 1 U 2 H p p r2 p r 2 and V2 Hp p r2 p r 2 form bases for N r k 1 k Nl 1 N r B k 2 and N l B k 2 respectively Denote 1 = [ U1 V 1 0 [ B U2 B 1 = V 2 0 [ C 0 C 1 = 0 0 If C R r G 1 G 2 B and C R l G 1 G 2 B then restricted matrix equation has a unique solution XB = C R r X R r k 1 Nr X N r B k 2 R l X R l k 1 Nl X N r B k 2 X = D CB D

14 Volume 26 pp 1-14 January Characterization of the W-Weighted Drazin Inverse Over the Quaternion Skew Field 13 and has the determinantal representations Suppose that H n n Ind = 1 r = r 1 < n B H p p IndB = 1 rb = r 2 < p and U 1 H n n r1 n r 1 V1 H n n r1 n r 1 U 2 H p p r2 p r 2 and V2 form bases for N r N l N r B and N l B respectively Denote H p p r2 p r 2 1 = [ U1 V 1 0 [ B U2 B 1 = V 2 0 [ C 0 C 1 = 0 0 If C R r G 1 G 2 B and C R l G 1 G 2 B then the restricted matrix equation XB = CR r X R r N r X N r BR l X R l N l X N r B has a unique solution X = g CB g and has the determinantal representations Conclusions In this paper we have proved a characterization of the W- weighted Drazin inverse over the quaternion skew field with applications Moreover some determinantal representations of the W-weighted Drazin inverse of a quaternion matrix within the framework of a theory of the row and column determinants are derived s applications we have shown the representations of the unique solution to some restricted quaternion matrix equations cknowledgment The author is very much indebted to the anonymous referees for their constructive and valuable comments and suggestions which greatly improve the original manuscript of this paper REFERENCES [1 SL Campbell and CD Meyer Generalized Inverses of Linear Transformations Pitman London 1979; Dover New York 1991 [2 N Castro Gonzáleza JJ Koliha and Y Wei Error bounds for perturbation of the Drazin inverse of closed operators with equal spectral projections ppl nal 81: [3 Y Wei and G Wang The perturbation theory for the Drazin inverse and its applications Linear lgebra ppl 258: [4 Y Wei and H Wu Challenging problems on the perturbation of Drazin inverse nn Oper Res 103: [5 RE Cline and TNE Greville Drazin inverse for rectangular matrices Linear lgebra ppl 29: [6 G Wang and J Sun Cramer rule for solution of the general restricted matrix equation ppl Math Comput 154: [7 G Wang and Y Wei The perturbation theory for the W-weighted Drazin inverse and its applications J Shanghai Teachers Univ Natural Sci 26:

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