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On the Hermitian-generalized Hamiltonian solutions of linear matrix equations. (English) Zbl 1087.15021
The authors give necessary and sufficient conditions for the solvability of the matrix equation $$AD=B$$ (with given $$n\times m$$ complex matrices $$D$$, $$B$$ and unknown $$n\times n$$-matrix $$A$$) in Hermitian-generalized Hamiltonian (HGH) matrices, i.e. in matrices $$A$$ satisfying the conditions $$A^*=A$$ and $$JAJ=A^*$$. Here $$A^*$$ is the transpose conjugate of $$A$$, and $$J$$ is a fixed real $$n\times n$$-matrix satisfying the conditions $$J^TJ=JJ^T=I$$ (the identity matrix) and $$J=-J^T$$. In the case of solvability the authors give the general representation of the solutions, and they solve the problem how to find the best approximation (shown to be unique) of a given complex $$n\times n$$-matrix by a HGH-solution to the above equation.

##### MSC:
 15A24 Matrix equations and identities
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