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Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B

  • * Corresponding author: Sihem Guerarra

    * Corresponding author: Sihem Guerarra
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  • This work is devoted to establish the extremal inertias ofthe two submatrices $X_{1}$ and $X_{4}$ in a Hermitian least rank solution $X$of the matrix equation $AXA^{*}=B$. From these formulas, necessary andsufficient conditions for these submatrices to be positive (nonpositive,negative, nonnegative) definite are achieved.

    Mathematics Subject Classification: Primary: 15A24; Secondary: 15A03, 15A09, 15B57.

    Citation:

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  •   S. L. Cambell and C. D. Meyer, Generalized Inverse of Linear Transformations, Society for Industrial and Applied Mathematics, 2008. doi: 10.1007/978-1-4612-0873-0.
      S. Guerarra  and  S. Guedjiba , Common Hermitian least-rank solution of matrix equations A1XA1* = B1 and A2XA2* = B2 subject to inequality restrictions, Facta universitatis (Niš). Ser. Math. Inform., 30 (2015) , 539-554.  doi: 10.2307/2152750.
      Y. Liu  and  Y. Tian , Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA* = B with applications, J. Appl. Math. Comput., 32 (2010) , 289-301.  doi: 10.2307/2152750.
      Y. Liu , Y. Tian  and  Y. Takane , Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA* = B, Linear Algebra Appl., 431 (2009) , 2359-2372.  doi: 10.2307/2152750.
      G. Marsaglia  and  G.P.H. Styan , Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra., 2 (1974) , 269-292.  doi: 10.2307/2152750.
      Y. Tian , The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math., 25 (2002) , 745-755.  doi: 10.2307/2152750.
      Y. Tian , Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 433 (2010) , 263-296.  doi: 10.2307/2152750.
      Y. Tian , Maximization and minimization of the rank and inertias of the Hermitian matrix expression A - BX - (BX)* with applications, Linear Algebra Appl., 434 (2011) , 2109-2139.  doi: 10.2307/2152750.
      Y. Tian , Least-squares solutions and least-rank solutions of the matrix equation AXA* = B and their relations, Numer. Linear Algebra Appl., 20 (2013) , 713-722.  doi: 10.2307/2152750.
      Y. Tian  and  S. Cheng , The maximal and minimal ranks of A - BXC with applications, New York Journal of Mathematics, 9 (2003) , 345-362.  doi: 10.2307/2152750.
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