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Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

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  • Predrag S. Stanimirović
  • Miroslav Ćirić
  • Igor Stojanović
  • Dimitrios Gerontitis

Abstract

Conditions for the existence and representations of -, -, and -inverses which satisfy certain conditions on ranges and/or null spaces are introduced. These representations are applicable to complex matrices and involve solutions of certain matrix equations. Algorithms arising from the introduced representations are developed. Particularly, these algorithms can be used to compute the Moore-Penrose inverse, the Drazin inverse, and the usual matrix inverse. The implementation of introduced algorithms is defined on the set of real matrices and it is based on the Simulink implementation of GNN models for solving the involved matrix equations. In this way, we develop computational procedures which generate various classes of inner and outer generalized inverses on the basis of resolving certain matrix equations. As a consequence, some new relationships between the problem of solving matrix equations and the problem of numerical computation of generalized inverses are established. Theoretical results are applicable to complex matrices and the developed algorithms are applicable to both the time-varying and time-invariant real matrices.

Suggested Citation

  • Predrag S. Stanimirović & Miroslav Ćirić & Igor Stojanović & Dimitrios Gerontitis, 2017. "Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses," Complexity, Hindawi, vol. 2017, pages 1-27, June.
    • Handle: RePEc:hin:complx:6429725
    DOI: 10.1155/2017/6429725
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    References listed on IDEAS

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    1. Xiaoji Liu & Yonghui Qin, 2012. "Successive Matrix Squaring Algorithm for Computing the Generalized Inverse," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-12, December.
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    Cited by:

    1. Kansal, Munish & Kumar, Sanjeev & Kaur, Manpreet, 2022. "An efficient matrix iteration family for finding the generalized outer inverse," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    2. Haifa Bin Jebreen, 2019. "Calculating the Weighted Moore–Penrose Inverse by a High Order Iteration Scheme," Mathematics, MDPI, vol. 7(8), pages 1-11, August.
    3. Stanimirović, Predrag S. & Petković, Marko D. & Mosić, Dijana, 2022. "Exact solutions and convergence of gradient based dynamical systems for computing outer inverses," Applied Mathematics and Computation, Elsevier, vol. 412(C).
    4. Dilan Ahmed & Mudhafar Hama & Karwan Hama Faraj Jwamer & Stanford Shateyi, 2019. "A Seventh-Order Scheme for Computing the Generalized Drazin Inverse," Mathematics, MDPI, vol. 7(7), pages 1-10, July.
    5. Stanimirović, Predrag S. & Ćirić, Miroslav & Lastra, Alberto & Sendra, Juan Rafael & Sendra, Juana, 2021. "Representations and symbolic computation of generalized inverses over fields," Applied Mathematics and Computation, Elsevier, vol. 406(C).
    6. Dijana Mosić & Predrag S. Stanimirović & Spyridon D. Mourtas, 2023. "Minimal Rank Properties of Outer Inverses with Prescribed Range and Null Space," Mathematics, MDPI, vol. 11(7), pages 1-18, April.
    7. Stanimirović, Predrag S. & Mosić, Dijana & Wei, Yimin, 2022. "Generalizations of composite inverses with certain image and/or kernel," Applied Mathematics and Computation, Elsevier, vol. 428(C).
    8. Ashim Kumar & Dijana Mosić & Predrag S. Stanimirović & Gurjinder Singh & Lev A. Kazakovtsev, 2022. "Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation," Mathematics, MDPI, vol. 10(15), pages 1-16, August.
    9. Chan, Eunice Y.S. & Corless, Robert M. & González-Vega, Laureano & Sendra, J. Rafael & Sendra, Juana, 2022. "Inner Bohemian inverses," Applied Mathematics and Computation, Elsevier, vol. 421(C).

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