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On solutions of the matrix equations $XF - AX = C$ and $XF - A\bar {X} =C$. (English) Zbl 1112.15018
The matrix equation $(*)\quad XF-AX=C$ is important in stability and control theory. When $F=A^*$, this is the Lyapunov matrix equation. The authors solve $(*)$ explicitly using the Kronecker map. The solution is expressed by a symmetric operator matrix, a controllability and an observability matrix. They express explicitly also the solution to the matrix equation $XF-A\bar{X}=C$ by means of the real representation of a complex matrix. The expression involves a symmetric operator matrix, two controllability and two observability matrices.

MSC:
15A24Matrix equations and identities
93B05Controllability
93B07Observability
93B25Algebraic theory of control systems
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References:
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