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Computational Aspects

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Generalized Inverses: Theory and Computations

Part of the book series: Developments in Mathematics ((DEVM,volume 53))

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Abstract

It follows from Chap. 1 that the six important kinds of generalized inverse: the M-P inverse \(A^\dag \), the weighted M-P inverse \(A_{MN}^{\dag }\), the group inverse \(A_g\), the Drazin inverse \(A_d\), the Bott-Duffin inverse \(A_{(L)}^{(-1)}\) and the generalized Bott-Duffin inverse \(A_{(L)}^{(\dag )}\) are all the generalized inverse \(A_{T,S}^{(2)}\), which is the \(\{ 2 \}\)-inverse of A with the prescribed range T and null space S.

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References

  1. B. Noble, J. Demmel, Applied Linear Algebra, 3rd edn. (Prentice-Hall, New Jersey, 1988)

    Google Scholar 

  2. G.W. Stewart, Introduction to Matrix Computation (Academic Press, New York, 1973)

    MATH  Google Scholar 

  3. G.H. Golub, C.F. Van Loan, Matrix Computations, 4th edn. (The Johns Hopkins University Press, Baltimore, MD, 2013)

    MATH  Google Scholar 

  4. C.F. Van Loan, Generalizing the singular value decomposition. SIAM J. Numer. Anal. 13, 76–83 (1976)

    Article  MathSciNet  Google Scholar 

  5. C.R. Rao, S.K. Mitra, Generalized Inverse of Matrices and Its Applications (John Wiley, New York, 1971)

    MATH  Google Scholar 

  6. S.L. Campbell, C.D. Meyer Jr., Generalized Inverses of Linear Transformations (Pitman, London, 1979)

    MATH  Google Scholar 

  7. C.D. Meyer, N.J. Rose, The index and the Drazin inverse of block triangular matrices. SIAM J. Appl. Math. 33, 1–7 (1977)

    Article  MathSciNet  Google Scholar 

  8. J. Miao, The Moore-Penrose inverse of a rank-\(r\) modified matrix. Numer. Math. J. Chinese Univ. 19, 355–361 (1989). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  9. J.M. Shoaf, The Drazin inverse of a rank-one modification of a square matrix. PhD thesis, North Carolina State University, 1975

    Google Scholar 

  10. Y. Wei, The weighted Moore-Penrose inverse of modified matrices. Appl. Math. Comput. 122, 1–13 (2001)

    MathSciNet  MATH  Google Scholar 

  11. Y. Wei, The Drazin inverse of a modified matrix. Appl. Math. Comput. 125, 295–301 (2002)

    MathSciNet  MATH  Google Scholar 

  12. T.N.E. Greville, Some applications of pseudoinverse of a martix. SIAM Rev. 2, 15–22 (1960)

    Article  MathSciNet  Google Scholar 

  13. A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, 2nd edn. (Springer, New York, 2003)

    MATH  Google Scholar 

  14. S. Dang, Matrix Theory and its Applications in Survey and Drawing (Survey and Drawing Press, 1980) (in Chinese)

    Google Scholar 

  15. X. He, W. Sun, The analysis of the Greville’s method. J. Nanjing Univ. 5, 1–10 (1988). in Chinese

    MATH  Google Scholar 

  16. G. Wang, A new proof of Greville method for computing the Moore-Penrose generalized inverse. J. Shanghai Normal Univ. 14, 32–38 (1985). in Chinese

    MATH  Google Scholar 

  17. G. Wang, Y. Chen, A recursive algorithm for computing the W-weighted Moore-Penrose inverse \(A_{MN}^{{\dagger }}\). J. Comput. Math. 4, 74–85 (1986)

    MathSciNet  Google Scholar 

  18. R.E. Cline, Representation of the generalized inverse of a partitioned matrix. J. Soc. Indust. Appl. Math. 12, 588–600 (1964)

    Article  MathSciNet  Google Scholar 

  19. L. Mihalyffy, An alternative representation of the generalized inverse of partitioned matrices. Linear Algebra Appl. 4, 95–100 (1971)

    Article  MathSciNet  Google Scholar 

  20. A. Ben-Israel, A note on partitioned matrices and equations. SIAM Rev. 11, 247–250 (1969)

    Article  MathSciNet  Google Scholar 

  21. J.V. Rao, Some more representations for generalized inverse of a partitioned matrix. SIAM J. Appl. Math. 24, 272–276 (1973)

    Article  MathSciNet  Google Scholar 

  22. J. Miao, Representations for the weighted Moore-Penrose inverse of a partitioned matrix. J. Comput. Math. 7, 320–323 (1989)

    MathSciNet  Google Scholar 

  23. J. Miao, Some results for computing the Drazin inverse of a partitioned matrix. J. Shanghai Normal Univ. 18, 25–31 (1989). in Chinese

    Google Scholar 

  24. R.E. Kalaba, N. Rasakhoo, Algorithm for generalized inverse. J. Optim. Theory Appl. 48, 427–435 (1986)

    Article  MathSciNet  Google Scholar 

  25. G. Wang, An imbedding method for computing the generalized inverse. J. Comput. Math. 8, 353–362 (1990)

    MathSciNet  MATH  Google Scholar 

  26. U.J. Le Verrier, Memoire sur les variations séculaires des éléments des orbites, pour les sept Planetes principales Mercure, Venus, La Terre, Mars, Jupiter (Bachelier, Saturne et Uranus, 1845)

    Google Scholar 

  27. D.K. Fadeev, V.N. Fadeeva, Computational Methods of Linear Algebra (W.H. Freeman & Co., Ltd, San Francisco, 1963)

    Google Scholar 

  28. M. Clique, J.G. Gille, On companion matrices and state variable feedback. Podstawy Sterowania 15, 367–376 (1985)

    MathSciNet  MATH  Google Scholar 

  29. H.P. Decell Jr., An application of the Cayley-Hamilton theorem to generalized matrix inversion. SIAM Rev. 7(4), 526–528 (1965)

    Article  MathSciNet  Google Scholar 

  30. R.E. Kalaba et al., A new proof for Decell’s finite algorithm for the generalized inverse. Appl. Math. Comput. 12, 199–211 (1983)

    MathSciNet  MATH  Google Scholar 

  31. G. Wang, A finite algorithm for computing the weighted Moore-Penrose inverse \(A_{MN}^{{\dagger }}\). Appl. Math. Comput. 23, 277–289 (1987)

    MathSciNet  Google Scholar 

  32. G. Wang, Y. Wei, Limiting expression for generalized inverse \(A_{T,S}^{(2)}\) and its corresponding projectors. Numer. Math., J. Chinese Univ. (English Series), 4, 25–30 (1995)

    Google Scholar 

  33. Y. Chen, Finite algorithm for the \((2)\)-generalized inverse \(A_{T, S}^{(2)}\). Linear Multilinear Algebra 40, 61–68 (1995)

    Article  MathSciNet  Google Scholar 

  34. G. Wang, Y. Lin, A new extension of Leverier’s algorithm. Linear Algebra Appl. 180, 227–238 (1993)

    Article  MathSciNet  Google Scholar 

  35. G. Wang, L. Qiu, Leverrier-Chebyshev algorithm for singular pencils. Linear Algebra Appl. 345, 1–8 (2002)

    Article  MathSciNet  Google Scholar 

  36. Z. Wu, B. Zheng, G. Wang, Leverrier-Chebyshev algorithm for the matrix polynomial of degree two. Numer. Math. J. Chinese Univ. (English Series), 11, 226–234 (2002)

    Google Scholar 

  37. Y. Chen, X. Shi, Y. Wei, Convergence of Rump’s method for computing the Moore-Penrose inverse. Czechoslovak Math. J., 66(141)(3), 859–879 (2016)

    Article  MathSciNet  Google Scholar 

  38. S. Miljković, M. Milandinović, P.S. Stanimirović, Y. Wei, Gradient methods for computing the Drazin-inverse solution. J. Comput. Appl. Math. 253, 255–263 (2013)

    Article  MathSciNet  Google Scholar 

  39. J. Miao, General expressions for the Moore-Penrose inverse of a \(2 \times 2\) block matrix. Linear Algebra Appl. 151, 1–15 (1991)

    Article  MathSciNet  Google Scholar 

  40. Y. Wei, Expression for the Drazin inverse of a \(2 \times 2\) block matrix. Linear Multilinear Algebra 45, 131–146 (1998)

    Article  MathSciNet  Google Scholar 

  41. S. Dang, A new method for computing the weighted generalized inverse of partitioned matrices. J. Comput. Math. 7, 324–326 (1989)

    MathSciNet  MATH  Google Scholar 

  42. J. Miao, The Moore-Penrose inverse of a rank-1 modified matrix. J. Shanghai Normal Univ. 17, 21–26 (1988). in Chinese

    Google Scholar 

  43. J. Miao, The Drazin inverse of Hessenberg matrices. J. Comput. Math. 8, 23–29 (1990)

    MATH  Google Scholar 

  44. Y. Wei, Y. Cao, H. Xiang, A note on the componentwise perturbation bounds of matrix inverse and linear systems. Appl. Math. Comput. 169, 1221–1236 (2005)

    MathSciNet  MATH  Google Scholar 

  45. J. Ji, The algebraic perturbation method for generalized inverses. J. Comput. Math. 7, 327–333 (1989)

    MathSciNet  MATH  Google Scholar 

  46. P.S. Stanimirović, Limit representations of generalized inverses and related methods. Appl. Math. Comput. 103, 51–68 (1999)

    MathSciNet  MATH  Google Scholar 

  47. J. Ji, An alternative limit expression of Drazin inverse and its applications. Appl. Math. Comput. 61, 151–156 (1994)

    MathSciNet  MATH  Google Scholar 

  48. S. Qiao, Recursive least squares algorithm for linear prediction problems. SIAM J. Matrix Anal. Appl. 9, 323–328 (1988)

    Article  MathSciNet  Google Scholar 

  49. S. Qiao, Fast adaptive RLS algorithms: a generalized inverse approach and analysis. IEEE Trans. Signal Process. 39, 1455–1459 (1991)

    Article  Google Scholar 

  50. G. Wang, Y. Wei, The iterative methods for computing the generalized inverse \(A_{MN}^{{\dagger }}\) and \(A_{d, W}\). Numer. Math. J. Chinese Univ., 16, 366–371 (1994). (in Chinese)

    Google Scholar 

  51. Y. Wei, H. Wu, (\(T\)-\(S\)) splitting methods for computing the generalized inverse \(A_{T, S}^{(2)}\) of rectangular systems. Int. J. Comput. Math. 77, 401–424 (2001)

    Article  MathSciNet  Google Scholar 

  52. Y. Wei, A characterization and representation of the generalized inverse \(A_{T, S}^{(2)}\) and its applications. Linear Algebra Appl. 280, 87–96 (1998)

    Article  MathSciNet  Google Scholar 

  53. Y. Wei, H. Wu, The representation and approximation for the generalized inverse \(A_{T, S}^{(2)}\). Appl. Math. Comput. 135, 263–276 (2003)

    MathSciNet  MATH  Google Scholar 

  54. X. Li, Y. Wei, A note on computing the generalized inverse \(A_{T, S}^{(2)}\) of a matrix \(A\). Int. J. Math. Math. Sci. 31, 497–507 (2002)

    Article  MathSciNet  Google Scholar 

  55. J.J. Climent, N. Thome, Y. Wei, A geometrical approach on generalized inverse by Neumann-type series. Linear Algebra Appl. 332–334, 535–542 (2001)

    MathSciNet  MATH  Google Scholar 

  56. X. Chen, W. Wang, Y. Song, Splitting based on the outer inverse of matrices. Appl. Math. Comput. 132, 353–368 (2002)

    MathSciNet  MATH  Google Scholar 

  57. X. Chen, G. Chen, A splitting method for the weighted Drazin inverse of rectangular matrices. J. East China Normal Univ. 8, 71–78 (1993). in Chinese

    MathSciNet  MATH  Google Scholar 

  58. X. Chen, R.E. Hartwig, The hyperpower iteration revisited. Linear Algebra Appl. 233, 207–229 (1996)

    Article  MathSciNet  Google Scholar 

  59. X. Liu, Y. Yu, J. Zhong, Y. Wei, Integral and limit representations of the outer inverse in Banach space. Linear Multilinear Algebra 60(3), 333–347 (2012)

    Article  MathSciNet  Google Scholar 

  60. E.D. Sontag, On generalized inverses of polynomial and other matrtices. IEEE Trans. Auto. Control AC 25, 514–517 (1980)

    Article  Google Scholar 

  61. J. Gao, G. Wang, Two algorithms for computing the Drazin inverse of a polynomial matrix. J. Shanghai Teach. Univ. (Natural Sciences) 31(2), 31–38 (2002). In Chinese

    Google Scholar 

  62. G. Wang, An application of the block-Cayley-Hamilton theorem. J. Shanghai Normal Univ. 20, 1–10 (1991). in Chinese

    MathSciNet  Google Scholar 

  63. Y. Wei, H. Wu, The representation and approximation for the Drazin inverse. J. Comput. Appl. Math. 126, 417–432 (2000)

    Article  MathSciNet  Google Scholar 

  64. Y. Wei, H. Wu, The representation and approximation for the weighted Moore-Penrose inverse. Appl. Math. Comput. 121, 17–28 (2001)

    MathSciNet  MATH  Google Scholar 

  65. Y. Wei, G. Wang, Approximate methods for the generalized inverse \(A_{T, S}^{(2)}\). J. Fudan Univ. 38, 233–240 (1999)

    MATH  Google Scholar 

  66. J. Wang, A recurrent neural networks for real-time matrix inversion. Appl. Math. Comput. 55, 23–34 (1993)

    Article  Google Scholar 

  67. J. Wang, Recurrent neural networks for computing pseudoinverse of rank-deficient matrices. SIAM J. Sci. Comput. 18, 1479–1493 (1997)

    Article  MathSciNet  Google Scholar 

  68. P.S. Stanimirović, I.S. Živković, Y. Wei, Neural network approach to computing outer inverses based on the full rank representation. Linear Algebra Appl. 501, 344–362 (2016)

    Article  MathSciNet  Google Scholar 

  69. P.S. Stanimirović, I.S. Živković, Y. Wei, Recurrent neural network for computing the Drazin inverse. IEEE Trans. Neural Netw. Learn. Syst. 26(11), 2830–2843 (2015)

    Article  MathSciNet  Google Scholar 

  70. X. Wang, H. Ma, P.S. Stanimirović, Recurrent neural network for computing the W-weighted Drazin inverse. Appl. Math. Comput. 300, 1–20 (2017)

    MathSciNet  Google Scholar 

  71. S. Qiao, X. Wang, Y. Wei, Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse. Linear Algebra Appl. 542, 101–117 (2018)

    Article  MathSciNet  Google Scholar 

  72. J. Jones, N. Karampetakis, A. Pugh, The computation and application of the generalized inverse via maple. J. Symbolic Comput. 25, 99–124 (1998)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, China

    Guorong Wang

  2. Department of Mathematics, Fudan University, Shanghai, China

    Yimin Wei

  3. Department of Computing and Software, McMaster University, Hamilton, ON, Canada

    Sanzheng Qiao

Authors
  1. Guorong Wang
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  2. Yimin Wei
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  3. Sanzheng Qiao
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Correspondence to Guorong Wang .

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Wang, G., Wei, Y., Qiao, S. (2018). Computational Aspects. In: Generalized Inverses: Theory and Computations. Developments in Mathematics, vol 53. Springer, Singapore. https://doi.org/10.1007/978-981-13-0146-9_5

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