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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.

15A24 Matrix equations and identities
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