Multiplicative Perturbation Bounds of the Group Inverse and Oblique Projection
|
|
- Sheila Smith
- 5 years ago
- Views:
Transcription
1 Filomat 30: 06, DOI 0.98/FIL67M Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: Multiplicative Perturbation Bounds of the Group Inverse Oblique Projection Lingsheng Meng a, Bing Zheng a a School of Mathematics Statistics, Lanzhou University, Lanzhou , PR China Abstract. In this paper, the multiplicative perturbation bounds of the group inverse related oblique projection under general unitarily invariant norm are presented by using the decompositions of B # A # BB # AA #.. Introduction Let C n n be the set of all n n complex matrices. For a given matrix A C n n, the symbols A, A, A, A #, IndA will st for the conjugate transpose, the spectral norm, general unitarily invariant norm, the group inverse, the index of A, respectively. I denotes the n n identity matrix. Also, for the sake of the simplicity in the later discussion, we will adopt the following notations with A C n n : P A = AA # = A # A, P A = I P A. In addition, we always assume that D D are n n nonsingular matrices throughout this paper. We recall that the group inverse A # of a matrix A C n n is the unique solution X, if exists, to the following three equations []: AXA = A, XAX = X, 3 AX = XA. A square matrix A has a group inverse if only if ranka = ranka, i.e., IndA =. When A # exists, P A = AA # is the projector which projects a vector on RA along NA, i.e., AA # = P RA,NA, where RA NA are the range null space of A, respectively. For the details of group inverse, we refer the readers to the book by Ben-Israel Greville []. The group inverse plays an important role in numerical analysis, Markov chains, etc. see [, 8, 9]. However, in most numerical applications the elements of A will seldom be known exactly, so it is necessary to know how its group inverse is perturbed when A is perturbed. There are extensive studies in this regard, e.g., [6, 9, 0], for the so-called additive perturbations, namely A is perturbed to B = A + E. Since the matrix scaling technique is often used to yield better-conditioned problems [3], the multiplicative perturbation model B = D AD with both D D are nonsingular matrices near the identities has been received much attention. Notice that B = D AD can be rewritten as B = A + E with E = I D A D AI D 00 Mathematics Subject Classification. Primary 5A09; Secondary 65F0 Keywords. Group inverse; multiplicative perturbation bound; oblique projection. Received: 0 September 04; Accepted: 9 November 04 Communicated by Dragana Cvetković Ilić Research supported by the National Natural Science Foundation of China No address: bzheng@lzu.edu.cn Bing Zheng
2 L. Meng, B. Zheng / Filomat 30: 06, or E = AI D I D AD, the multiplicative perturbation is a special additive perturbation. Hence the existing additive perturbation bounds can be applied directly to give the multiplicative perturbation bounds. But in general this technique may produce unideal perturbation bounds because it overlooks the nature of the multiplicative perturbation. Various multiplicative perturbation analysis have been done to many problems, such as the polar decomposition [4], the singular value decomposition [5], the Moore- Penrose inverse [7] when A is multiplicatively perturbed. In this paper, we will study the multiplicative perturbation bounds to the group inverse the related oblique projection under unitarily invariant norm.. Multiplicative Perturbation Bounds of the Oblique Projection In this section, we study the multiplicative perturbation bounds of the oblique projection related to group inverse under general unitarily invariant norm. Let A C n n, B = D AD IndA =, then B # may not exist see Example. After a simple analysis, we can get the following theorem. Theorem.. Let A C n n B = D AD, then B # exists if only if rankad D A = ranka. Example. Let A = 0 0 0, D = I D = 0 0 B = D AD = Then It is easy to see that ranka = ranka = 0 = rankb < rankb =. Hence, B # does not exist. In order to get the multiplicative perturbation bounds of the oblique projection, we fist give a decomposition of BB # AA #. Lemma.. Let A C n n, B = D AD IndA =. If rankad D A = ranka, then P B P A = P B I D P A + P B D IP A.. Proof. Since P B P A = B # B AI AA # + I BB # B AA #. B A = BI D + D IA,.3 combining..3, we immediately get.. Based on the above lemma, we can give a multiplicative perturbation bound for P B P A. But before stating this result, we mention another lemma which will be used. Lemma.3. [0] Let A C n n with IndA =. Then AA # = I AA #. Theorem.4. Let A C n n, B = D AD IndA =. If rankad D A = ranka, then P B P A P A P B I D + I D..4 If the used norm is the spectral norm, we have. P B P A P A P B I D + I D..5
3 L. Meng, B. Zheng / Filomat 30: 06, Proof. Note that the spectral norm is a special unitarily invariant norm.5 is a direct consequence of.4. So we only prove.4. Taking the unitarily invariant norm on both sides of. using Lemma., we get the estimate P B P A P B P A I D + P A P B I D = P A P B I D + I D. The proof is completed. Corollary.5. Under the assumptions of Theorem.4, if rankad D A = ranka γ = P A I D + I D <, then we have P A + γ P B P A γ.6 i.e., P B P A P A γ I D + I D..7 Proof. From.4, we have P B P A P B P A P A P B I D + I D, γ P B P A, since γ > 0, we get the right inequality of.6. Similarly, from P A P B P B P A, we can get the left inequality of.6. It is obvious that,.7 is the direct consequence of Multiplicative Perturbation Bounds of the Group Inverse In this section, we will provide multiplicative perturbation bounds of the group inverse under general unitarily invariant norm. To this end, we need the following lemma. Lemma 3.. Let A C n n, B = D AD such that IndA = rankad D A = ranka, then we have B # = P B D A# D P B 3. = I + Θ D, D A # I + Θ D, D, 3. where Θ D, D = P B D I P BI D, Θ D, D = D I P B I D P B D denotes the inverse of the conjugate transpose of D. Proof. Let Z be the matrix on the right side of equation 3.. From B = D AD, we have BZ = BD A# D BB# = D AA# AD B # = BB # ZB = B # BD A# D B = B# D AD = B # B. Hence, BZB = B, ZBZ = Z BZ = ZB, i.e., B # = Z. To prove 3., it is only needed to prove I + Θ D, D A # = P B D A# A # I + Θ D, D = A # D P B.
4 L. Meng, B. Zheng / Filomat 30: 06, In fact, we have I + Θ D, D A # = [I + I B # BD I B# BI D ]AA# A # = A # I B # BA # B # BI D A# = P B D A#. Similarly, we can prove A # I + Θ D, D = A # D P B. Obviously, Lemma 3. is valid both for full rank rank deficient matrices. Combining.6 3., we can get the following corollary. Corollary 3.. With the same assumptions as in Lemma 3.. max{ I D, I D } <, we have If γ = P A I D + I D < B # P A γ ΦD, D A#, where ΦD, D = I D I D. The following corollary presents an expression for B # A # that follows directly from 3., which is very important to get the perturbation bound of group inverse. Corollary 3.3. Let Θ D, D Θ D, D be the same as in Lemma 3.. With the same assumptions as in Lemma 3., we have B # A # = Θ D, D A # + A # Θ D, D + Θ D, D A # Θ D, D. 3.3 Next, we state the main result in this section. Theorem 3.4. With the same assumptions as in Lemma 3., then B # A # A # [ P B I D + I D + I D + I D + P B I D + I D I D + I D ]. 3.4 If γ = P A I D + I D <, we have B # A # A # [ P A γ I D + I D + I D + I D Proof. From Lemma., we can get + P A γ I D + I D I D + I D ]. 3.5 Θ D, D P B I D + I D 3.6 Θ D, D P B I D + I D. 3.7 Hence, 3.4 follows directly from 3.3, Substituting.6 into 3.4, we can get 3.5 immediately.
5 L. Meng, B. Zheng / Filomat 30: 06, Numerical Comparison In this section, we give an example to illustrate that the multiplicative perturbation bound is much better, in some cases, than the additive perturbation bound. To compare the multiplicative perturbation bound with the additive perturbation bound, we first mention the additive perturbation bounds of group inverse related oblique projection which were obtained in [0]. Let A, B = A + E C n n IndA = IndB =, then B # A # + max{ P A, P B } max{ A #, B# } E 4. P B P A max{ P A, P B } max{ A #, B # } E Example. Let A = , D = 0 0 D = I 3. Then 0 0 B = D AD = E = B A = 0.0 The estimates of the additive multiplicative perturbation bounds of group inverse associated oblique projector can be found in the following table we choose the spectral norm: Exact value Additive bound 4. Multiplicative bound 3.4 B # A # Exact value Additive bound 4. Multiplicative bound.4 P B P A Obviously, the multiplicative perturbation bounds of the group inverse the oblique projection are much better than the additive perturbation bounds for the above-mentioned example.. References [] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory Applications, nd edition, Springer, New York, 003. [] P. Bhimasankaram, D. Carlson, The group inverse of bordered matrix its application, Technical Report, No. 57, University of California, Santa Barbara, CA, 988. [3] X.W. Chang, R.C. Li, Multipliative perturbation analysis for QR factorizations, Numer. Algebra Control Optim [4] R.C. Li, Relative perturbation bounds for the unitary polar factor, BIT [5] R.C. Li, Relative perturbation theory: I. eigenvalue singular value variations, SIAM J. Matrix Anal. Appl [6] X.Z. Li, Y. Wei, An improvement on the perturbation of the group inverse oblique projection, Linear Algebra Appl [7] L.S. Meng, B. Zheng, The optimal perturbation bounds of the Moore-Penrose inverse under the Frobenius norm, Linear Algebra Appl [8] Y. Wei, A characterization representation of the Drazin inverse, SIAM J. Matrix Anal. Appl [9] Y. Wei, G. Wang, The perturbation theory for the Drazin inverse its applications, Linear Algebra Appl [0] Y. Wei, On the perturbation of the group inverse oblique projection, Appl. Math. Comp
The DMP Inverse for Rectangular Matrices
Filomat 31:19 (2017, 6015 6019 https://doi.org/10.2298/fil1719015m Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://.pmf.ni.ac.rs/filomat The DMP Inverse for
More informationELA THE OPTIMAL PERTURBATION BOUNDS FOR THE WEIGHTED MOORE-PENROSE INVERSE. 1. Introduction. Let C m n be the set of complex m n matrices and C m n
Electronic Journal of Linear Algebra ISSN 08-380 Volume 22, pp. 52-538, May 20 THE OPTIMAL PERTURBATION BOUNDS FOR THE WEIGHTED MOORE-PENROSE INVERSE WEI-WEI XU, LI-XIA CAI, AND WEN LI Abstract. In this
More informationSome results on the reverse order law in rings with involution
Some results on the reverse order law in rings with involution Dijana Mosić and Dragan S. Djordjević Abstract We investigate some necessary and sufficient conditions for the hybrid reverse order law (ab)
More informationGroup inverse for the block matrix with two identical subblocks over skew fields
Electronic Journal of Linear Algebra Volume 21 Volume 21 2010 Article 7 2010 Group inverse for the block matrix with two identical subblocks over skew fields Jiemei Zhao Changjiang Bu Follow this and additional
More informationRe-nnd solutions of the matrix equation AXB = C
Re-nnd solutions of the matrix equation AXB = C Dragana S. Cvetković-Ilić Abstract In this article we consider Re-nnd solutions of the equation AXB = C with respect to X, where A, B, C are given matrices.
More informationWorkshop on Generalized Inverse and its Applications. Invited Speakers Program Abstract
Workshop on Generalized Inverse and its Applications Invited Speakers Program Abstract Southeast University, Nanjing November 2-4, 2012 Invited Speakers: Changjiang Bu, College of Science, Harbin Engineering
More informationELA ON THE GROUP INVERSE OF LINEAR COMBINATIONS OF TWO GROUP INVERTIBLE MATRICES
ON THE GROUP INVERSE OF LINEAR COMBINATIONS OF TWO GROUP INVERTIBLE MATRICES XIAOJI LIU, LINGLING WU, AND JULIO BENíTEZ Abstract. In this paper, some formulas are found for the group inverse of ap +bq,
More informationYimin Wei a,b,,1, Xiezhang Li c,2, Fanbin Bu d, Fuzhen Zhang e. Abstract
Linear Algebra and its Applications 49 (006) 765 77 wwwelseviercom/locate/laa Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices Application of perturbation theory
More informationChapter 1. Matrix Algebra
ST4233, Linear Models, Semester 1 2008-2009 Chapter 1. Matrix Algebra 1 Matrix and vector notation Definition 1.1 A matrix is a rectangular or square array of numbers of variables. We use uppercase boldface
More informationNonsingularity and group invertibility of linear combinations of two k-potent matrices
Nonsingularity and group invertibility of linear combinations of two k-potent matrices Julio Benítez a Xiaoji Liu b Tongping Zhu c a Departamento de Matemática Aplicada, Instituto de Matemática Multidisciplinar,
More informationGeneralized Principal Pivot Transform
Generalized Principal Pivot Transform M. Rajesh Kannan and R. B. Bapat Indian Statistical Institute New Delhi, 110016, India Abstract The generalized principal pivot transform is a generalization of the
More informationSingular Value Inequalities for Real and Imaginary Parts of Matrices
Filomat 3:1 16, 63 69 DOI 1.98/FIL16163C Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Singular Value Inequalities for Real Imaginary
More informationσ 11 σ 22 σ pp 0 with p = min(n, m) The σ ii s are the singular values. Notation change σ ii A 1 σ 2
HE SINGULAR VALUE DECOMPOSIION he SVD existence - properties. Pseudo-inverses and the SVD Use of SVD for least-squares problems Applications of the SVD he Singular Value Decomposition (SVD) heorem For
More informationADDITIVE RESULTS FOR THE GENERALIZED DRAZIN INVERSE
ADDITIVE RESULTS FOR THE GENERALIZED DRAZIN INVERSE Dragan S Djordjević and Yimin Wei Abstract Additive perturbation results for the generalized Drazin inverse of Banach space operators are presented Precisely
More informationOperators with Compatible Ranges
Filomat : (7), 579 585 https://doiorg/98/fil7579d Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat Operators with Compatible Ranges
More informationA Note on the Group Inverses of Block Matrices Over Rings
Filomat 31:1 017, 3685 369 https://doiorg/1098/fil171685z Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat A Note on the Group Inverses
More informationEXPLICIT SOLUTION OF THE OPERATOR EQUATION A X + X A = B
EXPLICIT SOLUTION OF THE OPERATOR EQUATION A X + X A = B Dragan S. Djordjević November 15, 2005 Abstract In this paper we find the explicit solution of the equation A X + X A = B for linear bounded operators
More informationarxiv: v1 [math.na] 1 Sep 2018
On the perturbation of an L -orthogonal projection Xuefeng Xu arxiv:18090000v1 [mathna] 1 Sep 018 September 5 018 Abstract The L -orthogonal projection is an important mathematical tool in scientific computing
More informationOptimization problems on the rank and inertia of the Hermitian matrix expression A BX (BX) with applications
Optimization problems on the rank and inertia of the Hermitian matrix expression A BX (BX) with applications Yongge Tian China Economics and Management Academy, Central University of Finance and Economics,
More informationA revisit to a reverse-order law for generalized inverses of a matrix product and its variations
A revisit to a reverse-order law for generalized inverses of a matrix product and its variations Yongge Tian CEMA, Central University of Finance and Economics, Beijing 100081, China Abstract. For a pair
More informationOn EP elements, normal elements and partial isometries in rings with involution
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012 Article 39 2012 On EP elements, normal elements and partial isometries in rings with involution Weixing Chen wxchen5888@163.com Follow this
More informationAPPLICATIONS OF THE HYPER-POWER METHOD FOR COMPUTING MATRIX PRODUCTS
Univ. Beograd. Publ. Eletrotehn. Fa. Ser. Mat. 15 (2004), 13 25. Available electronically at http: //matematia.etf.bg.ac.yu APPLICATIONS OF THE HYPER-POWER METHOD FOR COMPUTING MATRIX PRODUCTS Predrag
More informationWeaker assumptions for convergence of extended block Kaczmarz and Jacobi projection algorithms
DOI: 10.1515/auom-2017-0004 An. Şt. Univ. Ovidius Constanţa Vol. 25(1),2017, 49 60 Weaker assumptions for convergence of extended block Kaczmarz and Jacobi projection algorithms Doina Carp, Ioana Pomparău,
More informationOn pairs of generalized and hypergeneralized projections on a Hilbert space
On pairs of generalized and hypergeneralized projections on a Hilbert space Sonja Radosavljević and Dragan S Djordjević February 16 01 Abstract In this paper we characterize generalized and hypergeneralized
More informationDragan S. Djordjević. 1. Introduction
UNIFIED APPROACH TO THE REVERSE ORDER RULE FOR GENERALIZED INVERSES Dragan S Djordjević Abstract In this paper we consider the reverse order rule of the form (AB) (2) KL = B(2) TS A(2) MN for outer generalized
More informationMoore Penrose inverses and commuting elements of C -algebras
Moore Penrose inverses and commuting elements of C -algebras Julio Benítez Abstract Let a be an element of a C -algebra A satisfying aa = a a, where a is the Moore Penrose inverse of a and let b A. We
More informationThe Drazin inverses of products and differences of orthogonal projections
J Math Anal Appl 335 7 64 71 wwwelseviercom/locate/jmaa The Drazin inverses of products and differences of orthogonal projections Chun Yuan Deng School of Mathematics Science, South China Normal University,
More informationELA THE MINIMUM-NORM LEAST-SQUARES SOLUTION OF A LINEAR SYSTEM AND SYMMETRIC RANK-ONE UPDATES
Volume 22, pp. 480-489, May 20 THE MINIMUM-NORM LEAST-SQUARES SOLUTION OF A LINEAR SYSTEM AND SYMMETRIC RANK-ONE UPDATES XUZHOU CHEN AND JUN JI Abstract. In this paper, we study the Moore-Penrose inverse
More informationDrazin Invertibility of Product and Difference of Idempotents in a Ring
Filomat 28:6 (2014), 1133 1137 DOI 10.2298/FIL1406133C Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Drazin Invertibility of
More informationThe characterizations and representations for the generalized inverses with prescribed idempotents in Banach algebras
Filomat 27:5 (2013), 851 863 DOI 10.2298/FIL1305851C Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat The characterizations and
More informationSOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES. S. S. Dragomir
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 5: 011), 151 16 DOI: 10.98/FIL110151D SOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR
More informationa Λ q 1. Introduction
International Journal of Pure and Applied Mathematics Volume 9 No 26, 959-97 ISSN: -88 (printed version); ISSN: -95 (on-line version) url: http://wwwijpameu doi: 272/ijpamv9i7 PAijpameu EXPLICI MOORE-PENROSE
More informationMoore-Penrose-invertible normal and Hermitian elements in rings
Moore-Penrose-invertible normal and Hermitian elements in rings Dijana Mosić and Dragan S. Djordjević Abstract In this paper we present several new characterizations of normal and Hermitian elements in
More informationApplied Mathematics Letters. Comparison theorems for a subclass of proper splittings of matrices
Applied Mathematics Letters 25 (202) 2339 2343 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Comparison theorems for a subclass
More informationOn the Moore-Penrose and the Drazin inverse of two projections on Hilbert space
On the Moore-Penrose and the Drazin inverse of two projections on Hilbert space Sonja Radosavljević and Dragan SDjordjević March 13, 2012 Abstract For two given orthogonal, generalized or hypergeneralized
More informationThe Residual Spectrum and the Continuous Spectrum of Upper Triangular Operator Matrices
Filomat 28:1 (2014, 65 71 DOI 10.2298/FIL1401065H Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat The Residual Spectrum and the
More informationarxiv: v1 [math.ra] 14 Apr 2018
Three it representations of the core-ep inverse Mengmeng Zhou a, Jianlong Chen b,, Tingting Li c, Dingguo Wang d arxiv:180.006v1 [math.ra] 1 Apr 018 a School of Mathematics, Southeast University, Nanjing,
More informationDiagonal and Monomial Solutions of the Matrix Equation AXB = C
Iranian Journal of Mathematical Sciences and Informatics Vol. 9, No. 1 (2014), pp 31-42 Diagonal and Monomial Solutions of the Matrix Equation AXB = C Massoud Aman Department of Mathematics, Faculty of
More informationELA
Electronic Journal of Linear Algebra ISSN 181-81 A publication of te International Linear Algebra Society ttp://mat.tecnion.ac.il/iic/ela RANK AND INERTIA OF SUBMATRICES OF THE MOORE PENROSE INVERSE OF
More informationSome results about the index of matrix and Drazin inverse
Mathematical Sciences Vol. 4, No. 3 (2010) 283-294 Some results about the index of matrix and Drazin inverse M. Nikuie a,1, M.K. Mirnia b, Y. Mahmoudi b a Member of Young Researchers Club, Islamic Azad
More informationFormulas for the Drazin Inverse of Matrices over Skew Fields
Filomat 30:12 2016 3377 3388 DOI 102298/FIL1612377S Published by Faculty of Sciences and Mathematics University of Niš Serbia Available at: http://wwwpmfniacrs/filomat Formulas for the Drazin Inverse of
More informationExpressions and Perturbations for the Moore Penrose Inverse of Bounded Linear Operators in Hilbert Spaces
Filomat 30:8 (016), 155 164 DOI 10.98/FIL1608155D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Expressions and Perturbations
More informationIntroduction to Numerical Linear Algebra II
Introduction to Numerical Linear Algebra II Petros Drineas These slides were prepared by Ilse Ipsen for the 2015 Gene Golub SIAM Summer School on RandNLA 1 / 49 Overview We will cover this material in
More informationA Note on Solutions of the Matrix Equation AXB = C
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER A: APPL MATH INFORM AND MECH vol 6, 1 (214), 45-55 A Note on Solutions of the Matrix Equation AXB C I V Jovović, B J Malešević Abstract:
More informationOn V-orthogonal projectors associated with a semi-norm
On V-orthogonal projectors associated with a semi-norm Short Title: V-orthogonal projectors Yongge Tian a, Yoshio Takane b a School of Economics, Shanghai University of Finance and Economics, Shanghai
More informationGeneralized left and right Weyl spectra of upper triangular operator matrices
Electronic Journal of Linear Algebra Volume 32 Volume 32 (2017 Article 3 2017 Generalized left and right Weyl spectra of upper triangular operator matrices Guojun ai 3695946@163.com Dragana S. Cvetkovic-Ilic
More informationarxiv: v1 [math.ra] 21 Jul 2013
Projections and Idempotents in -reducing Rings Involving the Moore-Penrose Inverse arxiv:1307.5528v1 [math.ra] 21 Jul 2013 Xiaoxiang Zhang, Shuangshuang Zhang, Jianlong Chen, Long Wang Department of Mathematics,
More informationWeighted Drazin inverse of a modified matrix
Functional Analysis, Approximation Computation 6 (2) (2014), 23 28 Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac Weighted Drazin inverse
More informationMatrix Inequalities by Means of Block Matrices 1
Mathematical Inequalities & Applications, Vol. 4, No. 4, 200, pp. 48-490. Matrix Inequalities by Means of Block Matrices Fuzhen Zhang 2 Department of Math, Science and Technology Nova Southeastern University,
More informationAdditive results for the generalized Drazin inverse in a Banach algebra
Additive results for the generalized Drazin inverse in a Banach algebra Dragana S. Cvetković-Ilić Dragan S. Djordjević and Yimin Wei* Abstract In this aer we investigate additive roerties of the generalized
More informationMOORE-PENROSE INVERSE IN AN INDEFINITE INNER PRODUCT SPACE
J. Appl. Math. & Computing Vol. 19(2005), No. 1-2, pp. 297-310 MOORE-PENROSE INVERSE IN AN INDEFINITE INNER PRODUCT SPACE K. KAMARAJ AND K. C. SIVAKUMAR Abstract. The concept of the Moore-Penrose inverse
More informationof a Two-Operator Product 1
Applied Mathematical Sciences, Vol. 7, 2013, no. 130, 6465-6474 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.39501 Reverse Order Law for {1, 3}-Inverse of a Two-Operator Product 1 XUE
More informationRank and inertia of submatrices of the Moore- Penrose inverse of a Hermitian matrix
Electronic Journal of Linear Algebra Volume 2 Volume 2 (21) Article 17 21 Rank and inertia of submatrices of te Moore- Penrose inverse of a Hermitian matrix Yongge Tian yongge.tian@gmail.com Follow tis
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 432 21 1691 172 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Drazin inverse of partitioned
More informationMinkowski Inverse for the Range Symmetric Block Matrix with Two Identical Sub-blocks Over Skew Fields in Minkowski Space M
International Journal of Mathematics And its Applications Volume 4, Issue 4 (2016), 67 80. ISSN: 2347-1557 Available Online: http://ijmaa.in/ International Journal 2347-1557 of Mathematics Applications
More information~ g-inverses are indeed an integral part of linear algebra and should be treated as such even at an elementary level.
Existence of Generalized Inverse: Ten Proofs and Some Remarks R B Bapat Introduction The theory of g-inverses has seen a substantial growth over the past few decades. It is an area of great theoretical
More informationSome additive results on Drazin inverse
Linear Algebra and its Applications 322 (2001) 207 217 www.elsevier.com/locate/laa Some additive results on Drazin inverse Robert E. Hartwig a, Guorong Wang a,b,1, Yimin Wei c,,2 a Mathematics Department,
More informationMATH36001 Generalized Inverses and the SVD 2015
MATH36001 Generalized Inverses and the SVD 201 1 Generalized Inverses of Matrices A matrix has an inverse only if it is square and nonsingular. However there are theoretical and practical applications
More informationApplied Matrix Algebra Lecture Notes Section 2.2. Gerald Höhn Department of Mathematics, Kansas State University
Applied Matrix Algebra Lecture Notes Section 22 Gerald Höhn Department of Mathematics, Kansas State University September, 216 Chapter 2 Matrices 22 Inverses Let (S) a 11 x 1 + a 12 x 2 + +a 1n x n = b
More informationEP elements in rings
EP elements in rings Dijana Mosić, Dragan S. Djordjević, J. J. Koliha Abstract In this paper we present a number of new characterizations of EP elements in rings with involution in purely algebraic terms,
More informationarxiv: v1 [math.ra] 25 Jul 2013
Representations of the Drazin inverse involving idempotents in a ring Huihui Zhu, Jianlong Chen arxiv:1307.6623v1 [math.ra] 25 Jul 2013 Abstract: We present some formulae for the Drazin inverse of difference
More informationarxiv: v1 [math.ra] 27 Jul 2013
Additive and product properties of Drazin inverses of elements in a ring arxiv:1307.7229v1 [math.ra] 27 Jul 2013 Huihui Zhu, Jianlong Chen Abstract: We study the Drazin inverses of the sum and product
More informationA note on solutions of linear systems
A note on solutions of linear systems Branko Malešević a, Ivana Jovović a, Milica Makragić a, Biljana Radičić b arxiv:134789v2 mathra 21 Jul 213 Abstract a Faculty of Electrical Engineering, University
More informationEP elements and Strongly Regular Rings
Filomat 32:1 (2018), 117 125 https://doi.org/10.2298/fil1801117y Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat EP elements and
More informationDecomposition of a ring induced by minus partial order
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012) Article 72 2012 Decomposition of a ring induced by minus partial order Dragan S Rakic rakicdragan@gmailcom Follow this and additional works
More informationDragan S. Djordjević. 1. Introduction and preliminaries
PRODUCTS OF EP OPERATORS ON HILBERT SPACES Dragan S. Djordjević Abstract. A Hilbert space operator A is called the EP operator, if the range of A is equal with the range of its adjoint A. In this article
More informationThe Hermitian R-symmetric Solutions of the Matrix Equation AXA = B
International Journal of Algebra, Vol. 6, 0, no. 9, 903-9 The Hermitian R-symmetric Solutions of the Matrix Equation AXA = B Qingfeng Xiao Department of Basic Dongguan olytechnic Dongguan 53808, China
More informationRepresentations for the Drazin Inverse of a Modified Matrix
Filomat 29:4 (2015), 853 863 DOI 10.2298/FIL1504853Z Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Representations for the Drazin
More informationRecurrent Neural Network Approach to Computation of Gener. Inverses
Recurrent Neural Network Approach to Computation of Generalized Inverses May 31, 2016 Introduction The problem of generalized inverses computation is closely related with the following Penrose equations:
More informationThe Expression for the Group Inverse of the Anti triangular Block Matrix
Filomat 28:6 2014, 1103 1112 DOI 102298/FIL1406103D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat The Expression for the Group Inverse
More informationLecture notes: Applied linear algebra Part 1. Version 2
Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
More informationWeak Monotonicity of Interval Matrices
Electronic Journal of Linear Algebra Volume 25 Volume 25 (2012) Article 9 2012 Weak Monotonicity of Interval Matrices Agarwal N. Sushama K. Premakumari K. C. Sivakumar kcskumar@iitm.ac.in Follow this and
More informationA Note on Simple Nonzero Finite Generalized Singular Values
A Note on Simple Nonzero Finite Generalized Singular Values Wei Ma Zheng-Jian Bai December 21 212 Abstract In this paper we study the sensitivity and second order perturbation expansions of simple nonzero
More informationSome Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces
Some Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces S.S. Dragomir Abstract. Some new inequalities for commutators that complement and in some instances improve recent results
More informationON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES
olume 10 2009, Issue 2, Article 41, 10 pp. ON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES HANYU LI, HU YANG, AND HUA SHAO COLLEGE OF MATHEMATICS AND PHYSICS CHONGQING UNIERSITY
More informationProduct of Range Symmetric Block Matrices in Minkowski Space
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 29(1) (2006), 59 68 Product of Range Symmetric Block Matrices in Minkowski Space 1
More informationPartial isometries and EP elements in rings with involution
Electronic Journal of Linear Algebra Volume 18 Volume 18 (2009) Article 55 2009 Partial isometries and EP elements in rings with involution Dijana Mosic dragan@pmf.ni.ac.yu Dragan S. Djordjevic Follow
More informationYongge Tian. China Economics and Management Academy, Central University of Finance and Economics, Beijing , China
On global optimizations of the rank and inertia of the matrix function A 1 B 1 XB 1 subject to a pair of matrix equations B 2 XB 2, B XB = A 2, A Yongge Tian China Economics and Management Academy, Central
More informationELA
REPRESENTATONS FOR THE DRAZN NVERSE OF BOUNDED OPERATORS ON BANACH SPACE DRAGANA S. CVETKOVĆ-LĆ AND MN WE Abstract. n this paper a representation is given for the Drazin inverse of a 2 2operator matrix
More informationAnalytical formulas for calculating the extremal ranks and inertias of A + BXB when X is a fixed-rank Hermitian matrix
Analytical formulas for calculating the extremal ranks and inertias of A + BXB when X is a fixed-rank Hermitian matrix Yongge Tian CEMA, Central University of Finance and Economics, Beijing 100081, China
More informationON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES
ON WEIGHTED PARTIAL ORDERINGS ON THE SET OF RECTANGULAR COMPLEX MATRICES HANYU LI, HU YANG College of Mathematics and Physics Chongqing University Chongqing, 400030, P.R. China EMail: lihy.hy@gmail.com,
More informationAnalytical formulas for calculating extremal ranks and inertias of quadratic matrix-valued functions and their applications
Analytical formulas for calculating extremal ranks and inertias of quadratic matrix-valued functions and their applications Yongge Tian CEMA, Central University of Finance and Economics, Beijing 100081,
More informationWe first repeat some well known facts about condition numbers for normwise and componentwise perturbations. Consider the matrix
BIT 39(1), pp. 143 151, 1999 ILL-CONDITIONEDNESS NEEDS NOT BE COMPONENTWISE NEAR TO ILL-POSEDNESS FOR LEAST SQUARES PROBLEMS SIEGFRIED M. RUMP Abstract. The condition number of a problem measures the sensitivity
More informationSTAT 309: MATHEMATICAL COMPUTATIONS I FALL 2017 LECTURE 5
STAT 39: MATHEMATICAL COMPUTATIONS I FALL 17 LECTURE 5 1 existence of svd Theorem 1 (Existence of SVD) Every matrix has a singular value decomposition (condensed version) Proof Let A C m n and for simplicity
More informationThe Moore-Penrose inverse of differences and products of projectors in a ring with involution
The Moore-Penrose inverse of differences and products of projectors in a ring with involution Huihui ZHU [1], Jianlong CHEN [1], Pedro PATRÍCIO [2] Abstract: In this paper, we study the Moore-Penrose inverses
More informationarxiv: v1 [math.ra] 16 Nov 2016
Vanishing Pseudo Schur Complements, Reverse Order Laws, Absorption Laws and Inheritance Properties Kavita Bisht arxiv:1611.05442v1 [math.ra] 16 Nov 2016 Department of Mathematics Indian Institute of Technology
More informationOn the Solution of Constrained and Weighted Linear Least Squares Problems
International Mathematical Forum, 1, 2006, no. 22, 1067-1076 On the Solution of Constrained and Weighted Linear Least Squares Problems Mohammedi R. Abdel-Aziz 1 Department of Mathematics and Computer Science
More informationarxiv: v1 [math.ra] 28 Jan 2016
The Moore-Penrose inverse in rings with involution arxiv:1601.07685v1 [math.ra] 28 Jan 2016 Sanzhang Xu and Jianlong Chen Department of Mathematics, Southeast University, Nanjing 210096, China Abstract:
More informationThe Solvability Conditions for the Inverse Eigenvalue Problem of Hermitian and Generalized Skew-Hamiltonian Matrices and Its Approximation
The Solvability Conditions for the Inverse Eigenvalue Problem of Hermitian and Generalized Skew-Hamiltonian Matrices and Its Approximation Zheng-jian Bai Abstract In this paper, we first consider the inverse
More informationSCHUR COMPLEMENTS IN C ALGEBRAS
mn header will be provided by the publisher SCHUR COMPLEMENTS IN C ALGEBRAS Dragana S. Cvetković-Ilić 1, Dragan S. Djordjević 1, and Vladimir Rakočević 1 1 Department of Mathematics, Faculty of Science
More informationSome results on matrix partial orderings and reverse order law
Electronic Journal of Linear Algebra Volume 20 Volume 20 2010 Article 19 2010 Some results on matrix partial orderings and reverse order law Julio Benitez jbenitez@mat.upv.es Xiaoji Liu Jin Zhong Follow
More informationConstruction of some Generalized Inverses of Operators between Banach Spaces and their Selections, Perturbations and Applications
Ph. D. Dissertation Construction of some Generalized Inverses of Operators between Banach Spaces and their Selections, Perturbations and Applications by Haifeng Ma Presented to the Faculty of Mathematics
More informationSome inequalities for sum and product of positive semide nite matrices
Linear Algebra and its Applications 293 (1999) 39±49 www.elsevier.com/locate/laa Some inequalities for sum and product of positive semide nite matrices Bo-Ying Wang a,1,2, Bo-Yan Xi a, Fuzhen Zhang b,
More informationTHE EXPLICIT EXPRESSION OF THE DRAZIN INVERSE OF SUMS OF TWO MATRICES AND ITS APPLICATION
italian journal of pure an applie mathematics n 33 04 (45 6) 45 THE EXPLICIT EXPRESSION OF THE DRAZIN INVERSE OF SUMS OF TWO MATRICES AND ITS APPLICATION Xiaoji Liu Liang Xu College of Science Guangxi
More informationLinGloss. A glossary of linear algebra
LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasi-triangular A matrix A is quasi-triangular iff it is a triangular matrix except its diagonal
More informationCharacterization of half-radial matrices
Characterization of half-radial matrices Iveta Hnětynková, Petr Tichý Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8, Czech Republic Abstract Numerical radius r(a) is the
More informationThe symmetric minimal rank solution of the matrix equation AX=B and the optimal approximation
Electronic Journal of Linear Algebra Volume 18 Volume 18 (2009 Article 23 2009 The symmetric minimal rank solution of the matrix equation AX=B and the optimal approximation Qing-feng Xiao qfxiao@hnu.cn
More informationRepresentation for the generalized Drazin inverse of block matrices in Banach algebras
Representation for the generalized Drazin inverse of block matrices in Banach algebras Dijana Mosić and Dragan S. Djordjević Abstract Several representations of the generalized Drazin inverse of a block
More informationMoore-Penrose Inverse of Product Operators in Hilbert C -Modules
Filomat 30:13 (2016), 3397 3402 DOI 10.2298/FIL1613397M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Moore-Penrose Inverse of
More informationMath Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p
Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the
More information