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Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA *=B with applications

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Abstract

Suppose that AXA *=B is a consistent matrix equation and partition its Hermitian solution X *=X into a 2-by-2 block form. In this paper, we give some formulas for the maximal and minimal ranks of the submatrices in an Hermitian solution X to AXA *=B. From these formulas we derive necessary and sufficient conditions for the submatrices to be zero or to be unique, respectively. As applications, we give some properties of Hermitian generalized inverses for an Hermitian matrix.

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Authors and Affiliations

  1. Department of Applied Mathematics, Shanghai Finance University, Shanghai, 201209, People’s Republic of China

    Yonghui Liu

  2. China Economics and Management Academy, Central University of Finance and Economics, Beijing, 100081, People’s Republic of China

    Yongge Tian

Authors
  1. Yonghui Liu
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  2. Yongge Tian
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Correspondence to Yonghui Liu.

Additional information

The work was supported by the Foundation of Shanghai Municipal Education Commission (07zz171) and the Leading Academic Discipline Project of Shanghai Municipal Education Commission (J51601).

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Liu, Y., Tian, Y. Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA *=B with applications. J. Appl. Math. Comput. 32, 289–301 (2010). https://doi.org/10.1007/s12190-009-0251-8

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  • DOI: https://doi.org/10.1007/s12190-009-0251-8

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