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A system of matrix equations and its applications

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Abstract

We derive the solvability conditions and an expression of the general solution to the system of matrix equations

$$ \begin{gathered} A_1 X = C_1 , \hfill \\ A_2 Y = C_2 ,YB_2 = D_2 ,Y = Y*, \hfill \\ A_3 Z = C_3 ,ZB_3 = D_3 ,Z = Z*, \hfill \\ B_2 X + (B_4 X)* + C_4 YC_{_4 }^* + D_4 ZD_{_4 }^* = A_4 . \hfill \\ \end{gathered} $$

. Moreover, we investigate the maximal and minimal ranks and inertias of Y and Z in the above system of matrix equations. As a special case of the results, we solve the problem proposed in Farid, Moslehian, Wang and Wu’s recent paper (Farid F O, Moslehian M S, Wang Q W, et al. On the Hermitian solutions to a system of adjointable operator equations. Linear Algebra Appl, 2012, 437: 1854–1891).

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    QingWen Wang & ZhuoHeng He

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  1. QingWen Wang
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  2. ZhuoHeng He
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Correspondence to QingWen Wang.

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Wang, Q., He, Z. A system of matrix equations and its applications. Sci. China Math. 56, 1795–1820 (2013). https://doi.org/10.1007/s11425-013-4596-y

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  • DOI: https://doi.org/10.1007/s11425-013-4596-y

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