Representations for the Drazin Inverse of a 2 x 2 Block Matrix

Two representations for the Drazin inverse of a $2\times2$ block matrix $M=[{A \atop C}\;{B \atop D}]$, where A and D are square matrices, in terms of the Drazin inverses of A and D have been recently developed under the assumptions that $C(I-AA^{D})=0$ and $(I-AA^{D})B=0$, and that the generalized Schur complement $D-CA^{D}B$ is either nonsingular or zero. These two representations of $M^{D}$ are extended to the case where $C(I-AA^{D})=0$ and $(I-AA^{D})B=0$ are substituted with $C(I-AA^{D})B=0$ and $A(I-AA^{D})B=0$. Moreover, upper bounds for the index of M are studied. Numerical examples are given to illustrate the new results.

  • [1]  Adi Ben‐Israel and , Thomas Greville, Generalized inverses: theory and applications, Robert E. Krieger Publishing Co. Inc., 1980xi+395, Corrected reprint of the 1974 original 81h:15005 Google Scholar

  • [2]  S. L. Campbell, Singular Systems of Differential Equations, Pitman, London, 1980. Google Scholar

  • [3]  S. L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear and Multilinear Algebra, 14 (1983), pp. 195–198. CrossrefGoogle Scholar

  • [4]  Chuan‐Kun Wu and , Ed Dawson, Existence of generalized inverse of linear transformations over finite fields, Finite Fields Appl., 4 (1998), 307–315 99i:15004 CrossrefGoogle Scholar

  • [5]  Xuzhou Chen and , Robert Hartwig, The group inverse of a triangular matrix, Linear Algebra Appl., 237/238 (1996), 97–108, Special issue honoring Calyampudi Radhakrishna Rao 10.1016/0024-3795(95)00561-7 97a:15008 CrossrefISIGoogle Scholar

  • [6]  Randall Cline and , T. Greville, A Drazin inverse for rectangular matrices, Linear Algebra Appl., 29 (1980), 53–62 10.1016/0024-3795(80)90230-X 81a:15007 CrossrefISIGoogle Scholar

  • [7]  Dragan Djordjević and , Predrag Stanimirović, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J., 51(126) (2001), 617–634 2002f:47001 CrossrefISIGoogle Scholar

  • [8]  R. Hartwig and , J. Shoaf, Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices, J. Austral. Math. Soc. Ser. A, 24 (1977), 10–34 57:3150 CrossrefGoogle Scholar

  • [9]  Robert Hartwig, The group inverse of a block triangular matrix, North‐Holland, New York, 1987, 137–146 88f:15007 Google Scholar

  • [10]  Robert Hartwig, Guorong Wang and , Yimin Wei, Some additive results on Drazin inverse, Linear Algebra Appl., 322 (2001), 207–217 10.1016/S0024-3795(00)00257-3 2001k:15004 CrossrefISIGoogle Scholar

  • [11]  Xiezhang Li and , Yimin Wei, An improvement on the perturbation of the group inverse and oblique projection, Linear Algebra Appl., 338 (2001), 53–66 2002h:15009 CrossrefISIGoogle Scholar

  • [12]  Jian Miao, Some results on the Drazin inverses of partitioned matrices, Shanghai Shifan Daxue Xuebao Ziran Kexue Ban, 18 (1989), 25–31 92d:15005 Google Scholar

  • [13]  Carl Meyer and , Nicholas Rose, The index and the Drazin inverse of block triangular matrices, SIAM J. Appl. Math., 33 (1977), 1–7 10.1137/0133001 57:345 LinkISIGoogle Scholar

  • [14]  M. Neumann and , H. Werner, Nonnegative group inverses, Linear Algebra Appl., 151 (1991), 85–96 10.1016/0024-3795(91)90355-Z 92e:15026 CrossrefISIGoogle Scholar

  • [15]  Yimin Wei, Expressions for the Drazin inverse of a 2×2 block matrix, Linear and Multilinear Algebra, 45 (1998), 131–146 99k:15007 CrossrefGoogle Scholar

  • [16]  Yimin Wei and , Xiezhang Li, An improvement on perturbation bounds for the Drazin inverse, Numer. Linear Algebra Appl., 10 (2003), 563–575, Dedicated to the 70th birthday of Ivo Marek 10.1002/nla.329 2004m:65052 CrossrefISIGoogle Scholar

  • [17]  Yimin Wei, Xiezhang Li and , Fanbin Bu, A perturbation bound of the Drazin inverse of a matrix by separation of simple invariant subspaces, SIAM J. Matrix Anal. Appl., 27 (2005), 72–81 MR2176807 LinkISIGoogle Scholar

  • [18]  Naimin Zhang and , Yimin Wei, Solving EP singular linear systems, Int. J. Comput. Math., 81 (2004), 1395–1405 MR2172927 CrossrefISIGoogle Scholar