Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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REPRESENTATIONS OF THE MOORE-PENROSE INVERSE OF 2×2 BLOCK OPERATOR VALUED MATRICES

  • Deng, Chun Yuan (COLLEGE OF MATHEMATICS SCIENCE SOUTH CHINA NORMAL UNIVERSITY) ;
  • Du, Hong Ke (COLLEGE OF MATHEMATICS AND INFORMATION SCIENCE SHAANXI NORMAL UNIVERSITY)
  • Published : 2009.11.01
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Abstract

We obtain necessary and sufficient conditions for $2{\tims}2$ block operator valued matrices to be Moore-Penrose (MP) invertible and give new representations of such MP inverses in terms of the individual blocks.

Keywords

References

  1. J. K. Baksalary and G. P. H. Styan, Generalized inverses of partitioned matrices in Banachiewicz-Schur form, Linear Algebra Appl. 354 (2002), no. 1-3, 41-47. https://doi.org/10.1016/S0024-3795(02)00334-8
  2. R. H. Bouldin, Generalized inverses and factorizations, Recent applications of generalized inverses, pp. 233-249, Res. Notes in Math., 66, Pitman, Boston, Mass.-London, 1982.
  3. R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-415. https://doi.org/10.2307/2035178
  4. M. R. Hestenes, Relative hermitian matrices, Pacific J. Math. 11 (1961), 225-245. https://doi.org/10.2140/pjm.1961.11.225
  5. J. Ji, Explicit expressions of the generalized inverses and condensed Cramer rules, Linear Algebra Appl. 404 (2005), 183-192. https://doi.org/10.1016/j.laa.2005.02.025
  6. M. Khadivi, Range inclusion and operator equations, J. Math. Anal. Appl. 197 (1996), no. 2, 630-633. https://doi.org/10.1006/jmaa.1996.0043
  7. T.-T. Lu and S.-H. Shiou, Inverses of 2 × 2 block matrices, Comput. Math. Appl. 43 (2002), no. 1-2, 119-129. https://doi.org/10.1016/S0898-1221(01)00278-4
  8. P. Phohomsiri and B. Han, An alternative proof for the recursive formulae for computing the Moore-Penrose M-inverse of a matrix, Appl. Math. Comput. 174 (2006), no. 1, 81-97. https://doi.org/10.1016/j.amc.2005.04.091
  9. Y. Tian, The Moore-Penrose inverses of m $\times$ n block matrices and their applications, Linear Algebra Appl. 283 (1998), no. 1-3, 35-60. https://doi.org/10.1016/S0024-3795(98)10049-6
  10. G. Wang and B. Zheng, The weighted generalized inverses of a partitioned matrix, Appl. Math. Comput. 155 (2004), no. 1, 221-233. https://doi.org/10.1016/S0096-3003(03)00772-0
  11. Y. Wei, The representation and approximation for the weighted Moore-Penrose inverse in Hilbert space, Appl. Math. Comput. 136 (2003), no. 2-3, 475-486. https://doi.org/10.1016/S0096-3003(02)00071-1
  12. Y. Wei, A characterization and representation of the generalized inverse $A_{T}^{(2)},_S$ and its applications, Linear Algebra Appl. 280 (1998), no. 2-3, 87-96. https://doi.org/10.1016/S0024-3795(98)00008-1
  13. Y. Wei, J. Cai, and M. K. Ng, Computing Moore-Penrose inverses of Toeplitz matrices by Newton's iteration, Math. Comput. Modelling 40 (2004), no. 1-2, 181-191. https://doi.org/10.1016/j.mcm.2003.09.036
  14. Y.Wei and J. Ding, Representations for Moore-Penrose inverses in Hilbert spaces, Appl. Math. Lett. 14 (2001), no. 5, 599-604. https://doi.org/10.1016/S0893-9659(00)00200-7
  15. Y. Wei and D. S. Djordjevi, On integral representation of the generalized inverse $A_{T}^{(2)},_S$, Appl. Math. Comput. 142 (2003), no. 1, 189-194. https://doi.org/10.1016/S0096-3003(02)00296-5
  16. Y. Wei and N. Zhang, A note on the representation and approximation of the outer inverse $A_{T}^{(2)},_S$ of a matrix A, Appl. Math. Comput. 147 (2004), no. 3, 837-841. https://doi.org/10.1016/S0096-3003(02)00815-9
  17. J. Zhow and G. Wang, Block idempotent matrices and generalized Schur complement, Appl. Math. Comput. 188 (2007), no. 1, 246-256. https://doi.org/10.1016/j.amc.2006.08.175

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