## 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.

### MSC:

 15A24 Matrix equations and identities 15A09 Theory of matrix inversion and generalized inverses 15A03 Vector spaces, linear dependence, rank, lineability
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