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The {P,Q,k+1}-reflexive solution of matrix equation AXB=C

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Abstract

Let P∈ℂm×m and Q∈ℂn×n be Hermitian and {k+1}-potent matrices, i.e., P k+1=P=P ∗, Q k+1=Q=Q ∗, where (⋅)∗ stands for the conjugate transpose of a matrix. A matrix X∈ℂm×n is called {P,Q,k+1}-reflexive (anti-reflexive) if PXQ=X (PXQ=−X). In this paper, the matrix equation AXB=C subject to {P,Q,k+1} reflexive and anti-reflexive constraints are studied by converting into two simpler cases: k=1 and k=2. The solvability conditions and the general solution are obtained, and then the maximal and minimal ranks among the solutions are derived. Finally, the associated optimal approximation problem for a given matrix is considered.

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Acknowledgements

The authors are grateful to Professor Sung Yell Song and two referees for their helpful comments and suggestions, which greatly improves the expressions and quality of this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu, 741001, P.R. China

    Mao-lin Liang, Li-fang Dai & Ya-fang Yang

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  1. Mao-lin Liang
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  2. Li-fang Dai
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  3. Ya-fang Yang
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Correspondence to Li-fang Dai.

Additional information

This work was supported by the research grants of Education Department of Gansu Province (No. 1108B-03) and Tianshui Normal University (No. TSA1104).

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Liang, Ml., Dai, Lf. & Yang, Yf. The {P,Q,k+1}-reflexive solution of matrix equation AXB=C . J. Appl. Math. Comput. 42, 339–350 (2013). https://doi.org/10.1007/s12190-012-0631-3

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  • DOI: https://doi.org/10.1007/s12190-012-0631-3

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