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**A system of matrix equations and its applications.**
*(English)*
Zbl 1291.15043

For the following system of matrix equations, \(A_1X = {C_1}\), \({A_2}Y = {C_2}\), \(Y{B_2} = {D_2}\), \(Y = {Y^ * }\), \({A_3}Z = {C_3}\), \(Z{B_3} = {D_3}\), \(Z = {Z^ * }\), \({B_4}X + {({B_4}X)^ * } + {C_4}YC_4^ * + {D_4}ZD_4^ * = {A_4}\), solvability conditions are proved, a general solution is formulated, and the maximal and minimal ranks and inertias of \(Y\) and \(Z\) are established. Finally, for the system \({A_2}Y = {C_2}\), \(Y{B_2} = {D_2}\), \({A_3}Z = {C_3}\), \(Z{B_3} = {D_3}\), \({C_4}YC_4^ * + {D_4}ZD_4^ * = {A_4}\), the maximal and minimal ranks and inertias of general Hermitian solutions are established and some necessary and sufficient conditions to have nonnegative definite, nonpositive definite, positive definite and negative definite solutions are proved.

Reviewer: Mihail Voicu (Iaşi)

### MSC:

15A24 | Matrix equations and identities |

15A09 | Theory of matrix inversion and generalized inverses |

15A03 | Vector spaces, linear dependence, rank, lineability |

### Keywords:

linear matrix equation; Moore-Penrose inverse; rank; inertia; positive definite solution; Hermitian solution
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\textit{Q. Wang} and \textit{Z. He}, Sci. China, Math. 56, No. 9, 1795--1820 (2013; Zbl 1291.15043)

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