By using the matrix decomposition and the reverse order law, we provide some new expressions of the Drazin inverse for any 2×2 block matrix with rank constraints

Linlin Zhao

Journal of Applied Mathematics

English

Directory of Open Access Journals

The Expression of the Drazin Inverse with Rank Constraints

By using the matrix decomposition and the reverse order law, we provide some new expressions of the Drazin inverse for any2×2block matrix with rank constraints.</jats:p

Crossref

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 390592, 10 pagesdoi:10.1155/2012/390592Research ArticleThe Expression of the Drazin Inverse withRank ConstraintsLinlin ZhaoDepartment of Mathematics, Dezhou University, Dezhou 253023, ChinaCorrespondence should be addressed to Linlin Zhao, zhaolinlin0635@163.comReceived 9 October 2012; Accepted 5 November 2012Academic Editor: C. ConcaCopyright q 2012 Linlin Zhao. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.By using the matrix decomposition and the reverse order law, we provide some new expressionsof the Drazin inverse for any 2 × 2 block matrix with rank constraints.1. IntroductionLet A be a square complex matrix. The symbols rA and A† stand for the rank and theMoore-Penrose inverse of thematrixA, respectively. The Drazin inverseAD ofA is the uniquematrix satisfyingAk1AD  Ak, ADAAD  AD, AAD  ADA, 1.1where k  indA is the index of A, the smallest nonnegative integer such that rAk1 rAk. We write Aπ  I −AAD.The Drazin inverse of a square matrix plays an important role in various fieldslike singular diﬀerential equations and singular diﬀerence equations, Markov chains, anditerative methods.The problem of finding explicit representations for the Drazin inverse of a complexblock matrix,M (A BC D)1.2in terms of its blocks was posed by Campbell and Meyer 1, 2 in 1979. Many authors haveconsidered this problem and have provided formulas forMD under some specific conditions3–6.2 Journal of Applied MathematicsIn this paper, under rank constraints, we will present some new representations ofMDwhich have not been discussed before.2. PreliminaryLemma 2.1 see 4. Let P and Q be square matrices of the same order.If PQ  0, thenP QD  Qπ[k−1∑i0Qi(PD)i]PD QD[k−1∑i0(QD)iP i]Pπ, 2.1wheremax{indP, indQ} ≤ k ≤ indP  indQ.If PQ  0 and QP  0, then P QD  PD QD.Lemma 2.2 see 7. LetM1 (A 0C B), M2 (B C0 A), 2.2where A, B are square matrices with indA  r, indB  s. ThenMD1 (AD 0X BD), MD2 (BD X0 AD), 2.3whereX (BD)2[r−1∑i0(BD)iCAi]Aπ  Bπ(s−1∑i0BiC(AD)i)(AD)2 − BDCAD. 2.4Lemma 2.3 see 8. Let A  A1A2 · · ·An, X  ADn ADn−1 · · ·AD1 . Then X  AD if and only ifA1, A2, . . . , An, A satisfyr(−1nA2k1 AkEETAk N) r(Ak) r(Ak1) · · ·  r(Akn), 2.5where Ai ∈ Cm×m, i  1, 2, . . . , n, E  0, . . . , 0, Im ∈ Cm×mn1 andN ⎛⎜⎜⎜⎜⎝A2k11 Ak1......A2k1n AknAkn−1Akn⎞⎟⎟⎟⎟⎠, 2.6with k  max{indAi, indA}, 1 ≤ i ≤ n.Journal of Applied Mathematics 3Lemma 2.4 see 9. Let A ∈ Cm×n, B ∈ Cm×k, C ∈ Cl×n. Thenr(A BC 0) rB  rC  r[(Im − BB†)A(In − C†C)]. 2.73. Main ResultsIn this section, with rank equality constraints, we consider the Drazin inverse of blockmatrices.Let M (A BC D), where A ∈ Cm×m is invertible and D − CA−1B ∈ Cn×n is singular. It iseasy to verify thatM can be decomposed asM (Im 0CA−1 In)(A 00 D − CA−1B)(Im A−1B0 In) M1M2M3. 3.1LetE  0, 0, 0, Imn, N ⎛⎜⎜⎜⎝M2k11 Mk1M2k12 Mk2Mk1M2k13 Mk3Mk2Mk3⎞⎟⎟⎟⎠, 3.2where k  max{indM2, indM}. According to Lemma 2.3, we have the followingtheorem.Theorem 3.1. LetM (A BC D), whereA ∈ Cm×m is invertible andD −CA−1B ∈ Cn×n is singular. Ifr(Mk) r(Mk2M−11 FMk 00 EMkM−13 Mk2) r(Mk2), 3.3where k  max{indM2, indM}, EMk  I − MkMk†, FMk  I − Mk†Mk, then MD hasthe following form:MD (AD A−1B(D − CA−1B)DCA−1 −A−1B(D − CA−1B)D−(D − CA−1B)DCA−1 (D − CA−1B)D). 3.4Proof. From Lemma 2.3 and 3.1, we know that ifr(−M2k1 MkEETMk N) r(Mk) r(Mk1) r(Mk2) r(Mk3), 3.54 Journal of Applied Mathematicswhere k  max{indM2, indM}, thenMD  M−13 MD2 M−11 (Im −A−1B0 In)(AD 00(D − CA−1B)D)(Im 0−CA−1 In)(AD A−1B(D − CA−1B)DCA−1 −A−1B(D − CA−1B)D−(D − CA−1B)DCA−1 (D − CA−1B)D).3.6Note thatr(−M2k1 MkEETMk N) r⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−M2k1 0 0 0 Mk0 0 0 M2k11 Mk10 0 M2k12 Mk2Mk1 00 M2k13 Mk3Mk2 0 0Mk Mk3 0 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ r⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0 Mk1Mk3 0 0 Mk0 0 0 M2k11 Mk10 0 M2k12 Mk2Mk1 00 M2k13 Mk3Mk2 0 0Mk Mk3 0 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ r⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0 0 0 0 Mk0 −Mk1MMk3 0 M2k11 Mk10 0 M2k12 Mk2Mk1 00 M2k13 Mk3Mk2 0 0Mk Mk3 0 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.3.7LetG (0, 0, 0,Mk), J ⎛⎜⎜⎝000Mk⎞⎟⎟⎠, H ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−Mk1MMk3 0 M2k11 Mk10 M2k12 Mk2Mk1 0M2k13 Mk3Mk2 0 0Mk3 0 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠. 3.8Journal of Applied Mathematics 5From Lemma 2.4, we haver(0 GJ H) r(H JG 0) rJ  rG  r[(I4m − JJ†)H(I4m −G†G)] 2r(Mk) r[(I4m − JJ†)H(I4m −G†G)].3.9Note that J†  0, 0, 0, Mk†, G† ( 000Mk†). Then we get(I4m − JJ†)H(I4m −G†G)⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−Mk1MMk3 0 M2k11 Mk1[I4m −(Mk)†Mk]0 M2k12 Mk2Mk1 0M2k13 Mk3Mk2 0 0[I4m −Mk(Mk)†]Mk3 0 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.3.10Let EMk  I4m −MkMk†, F  I4m − Mk†Mk. Then I4m − JJ†HI4m −G†G can berewritten as the following three matrix products:⎛⎜⎜⎝Mk1 0 0 00 Im 0 00 0 Mk3 00 0 0 Im⎞⎟⎟⎠⎛⎜⎜⎜⎝−M 0 M1 FkM0 M2k12 Mk2 0M3 Mk2 0 0EkM 0 0 0⎞⎟⎟⎟⎠⎛⎜⎜⎝Mk3 0 0 00 Im 0 00 0 Mk1 00 0 0 Im⎞⎟⎟⎠. 3.11SinceM1,M3 is nonsingular, thenr[(I4m − JJ†)H(I4m −G†G)] r⎛⎜⎜⎝−M 0 M1 FMk0 M2k12 Mk2 0M3 Mk2 0 0EMk 0 0 0⎞⎟⎟⎠ r⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−M 0 Imn FMk0 M2k12 Mk2M−11 0Imn M−13 Mk2 0 0EMk 0 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠6 Journal of Applied Mathematics r⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝0 M1Mk12 Imn FMk0 M2k12 Mk2M−11 0Imn M−13 Mk2 0 00 −EMkM−13 Mk2 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ r⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝0 M1Mk12 Imn FMk0 0 0 −Mk2M−11 FMkImn M−13 Mk2 0 00 −EMkM−13 Mk2 0 0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ r⎛⎜⎜⎝0 0 Imn 00 0 0 −Mk2M−11 FMkImn 0 0 00 −EMkM−13 Mk2 0 0⎞⎟⎟⎠ 2m  n  r(Mk2M−11 FMk 00 EMkM−13 Mk2).3.12Thus, we haver(−M2k1 MkEETMk N) r(0 GJ H) 2r(Mk) 2m  n  r(Mk2M−11 FMk 00 EMkM−13 Mk2).3.13From the above equality and the condition 3.3, 3.5 is easily verified.Let M (A BC D), where A ∈ Cm×m, D ∈ Cn×n. It is easy to verify that the matrix M canbe decomposed asM (Im 0CAD In)(A AπBCAπ S)(Im ADB0 In) A1A2A3, 3.14where S  D − CADB is the generalized Schur complement of A inM.Journal of Applied Mathematics 7LetN1 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝A2k11 Ak1A2k12 Ak2Ak1A2k13 Ak3Ak2Ak3⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠, 3.15where k  max {indA2, indM}. Then we have the following theorem.Theorem 3.2. If AAπB  0, CAπB  0, and the matrices A1, A2, A3,M satisfyr(Mk) r(Ak2A−11 FMk 00 EMkA−13 Ak2) r(Ak2), 3.16then,MD (AD AπBXAD (AπBSD −ADB)Y AπB(SD)2 −ADBSDY SD), 3.17where S  D − CADB, Y  SD2∑indA−1i0 SDiCAπAiAπ − SDCAD.Proof. From Lemma 2.3 and 3.14, we get that ifr(−M2k1 MkEETMk N1) r(Mk) r(Ak1) r(Ak2) r(Ak3), 3.18thenMD  A−13 AD2 A−11 (Im −ADB0 In)(A AπBCAπ S)D(Im 0−CAD In). 3.19Similar to the proof of Theorem 3.1, we derive that the rank condition 3.18 can besimplified as 3.16.Next, we will give the representation for AD2 . LetA2 (A 0CAπ S)(0 AπB0 0) P Q. 3.20Since AAπB  0, CAπB  0, and AπAπ  Aπ , then PQ  0. From Lemma 2.2, we getQD  0, PD (AD 0X SD), where X (SD)2[indA−1∑i0(SD)iCAπAi]Aπ. 3.218 Journal of Applied MathematicsFrom Lemma 2.1 and the fact Qi  0, i ≥ 2, it follows thatAD2  P QD (Imn −QQD)(Imn QPD)PD[Imn (0 AπB0 0)(AD 0X SD)](AD 0X SD)(Im AπBX AπBSD0 In)(AD 0X SD)(AD AπB(XAD  SDX)AπB(SD)2X SD).3.22Substituting AD2 in 3.19, the conclusion can be obtained.From Theorem 3.2, we can easily obtain the following corollaries.Corollary 3.3. If AπB  0 and the rank equality 3.16 hold, thenMD (AD −ADB(X − SDCAD) −ADBSDX − SDCAD SD), 3.23where S and X are the same as in Theorem 3.2.Corollary 3.4. If AπB  0, CAπ  0, and the rank equality 3.16 hold, thenMD (AD ADBSDCAD −ADBSD−SDCAD SD), 3.24where S  D − CADB.Next, we will consider another decomposition of M involving the generalized Schurcomplement SB  C −DBDA.LetM (A BC D), where A,D ∈ Cm×m. ThenM can be decomposed asM (Im 0DBD Im)(BπA BSB DBπ)(Im 0BDA Im) V1V2V3, 3.25where SB  C −DBDA is the generalized Schur complement of B inM.LetE1  0, 0, 0, I2m, N2 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝V 2k11 Vk1V 2k12 Vk2 Vk1V 2k13 Vk3 Vk2V k3⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠, 3.26where k  max{indV2, indM}. Then we have the following theorem.Journal of Applied Mathematics 9Theorem 3.5. Let M (A BC D), where A, D ∈ Cm×m. If BπA  0, CB  DBDAB and the matricesV1, V2, V3,M satisfy the following rank equality:r(Mk) r(V k2 V−11 FMk 00 EMkV −13 Vk2) r(V k2), 3.27thenMD (BX2(XSB −DBD)BX2(I − BDABX)X(XSB −DBD) (I − BDABX)X), 3.28where X  DBπD.Proof. From Lemma 2.3 and 3.25, we get that if the following rank conditionr(−M2k1 MkEETMk N2) r(Mk) r(V k1) r(V k2) r(V k3)3.29holds, then M  V −13 VD2 V−11 . From the same method used in Theorem 3.1, we can verify thatthe above condition 3.29 can be reduced to 3.27.Next, we will give the representation for VD2 . For BπA  0, we writeV2 (0 0SB DBπ)(0 B0 0) P1 Q1, 3.30where SB  C −DBDA.From the condition CB  DBDAB, we get P1Q1  0, according to Lemma 2.2, we haveQD1  0, PD1 ⎡⎣ 0 0(DBπD)2SB DBπD⎤⎦. 3.31Let X  DBπD. By Lemma 2.1 and the fact Qi1  0, i ≥ 2, we getVD2 (I2m −Q1QD1)(I2m Q1PD1)PD1[I2m (0 B0 0)(0 0X2SB X)](0 0X2SB X)(Im  BX2SB BX0 Im)(0 0X2SB X)(BX3SBX BX2X2SB X).3.3210 Journal of Applied MathematicsTherefore, we getMD (Im 0−BDA Im)(BX3SBX BX2X2SB X)(Im 0−DBD Im)(BX2(XSB −DBD)BX2(I − BDABX)X(XSB −DBD) (I − BDABX)X).3.33Remark 3.6. In addition to the decompositions of M in 3.14 and 3.25, the matrix M alsocan be decomposed as other matrix products involving the generalized Schur complementsSC  B −ACDD or SD  A − BDDC. In these cases, new formulas for MD would be derivedby the method used in this paper.4. ConclusionIn this paper, we mainly discuss the Drazin inverse of block matrices under rank equalityconstraints. Comparing with the existing results, it is obvious that our results have morestrong restrictions, but the methods used in this paper are diﬀerent from those in previousrelevant paper.AcknowledgmentsThe author wishes to thank Professor Guoliang Chen and Dr. Jing Cai for their help. Thisresearch is financed by NSFC Grants 10971070, 11071079, and 10901056.References1 S. L. Campbell and C. D. Meyer, GeneralizedInverses of Linear Transformations, Dover, New York, NY,USA, 1991.2 S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, UK,1979.3 Y. Wei, “Expressions for the Drazin inverse of a 2 × 2 block matrix,” Linear and Multilinear Algebra, vol.45, no. 2-3, pp. 131–146, 1998.4 R. E. Hartwig, G. Wang, and Y. Wei, “Some additive results on Drazin inverse,” Linear Algebra and ItsApplications, vol. 322, no. 1–3, pp. 207–217, 2001.5 X. Li and Y. Wei, “A note on the representations for the Drazin inverse of 2 × 2 block matrices,” LinearAlgebra and Its Applications, vol. 423, no. 2-3, pp. 332–338, 2007.6 D. S. Cvetkovic´-Ilic´, “A note on the representation for the Drazin inverse of 2×2 block matrices,” LinearAlgebra and Its Applications, vol. 429, no. 1, pp. 242–248, 2008.7 C. D. Meyer, and N. J. Rose, “The index and the Drazin inverse of block triangular matrices,” SIAMJournal on Applied Mathematics, vol. 33, no. 1, pp. 1–7, 1977.8 G. Wang, “The reverse order law for the Drazin inverses of multiple matrix products,” Linear Algebraand Its Applications, vol. 348, pp. 265–272, 2002.9 G. Marsaglia and G. P. H. Styan, “Equalities and inequalities for ranks of matrices,” Linear andMultilinear Algebra, vol. 2, pp. 269–292, 1974.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical Problems in EngineeringHindawi Publishing Corporationhttp://www.hindawi.comDifferential EquationsInternational Journal ofVolume 2014Applied MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical PhysicsAdvances inComplex AnalysisJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014OptimizationJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Operations ResearchAdvances inJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Function SpacesAbstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal of Mathematics and Mathematical SciencesHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AlgebraDiscrete Dynamics in Nature and SocietyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Decision SciencesAdvances inDiscrete MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Stochastic AnalysisInternational Journal of

http://doaj.org/search?source=%7B%22query%22%3A%7B%22bool%22%3A%7B%22must%22%3A%5B%7B%22term%22%3A%7B%22id%22%3A%22fb5955d307da4be592320381e485b734%22%7D%7D%5D%7D%7D%7D

The Expression of the Drazin Inverse with Rank Constraints

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals

Crossref