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Eigenvector-free solutions to \(AX = B\) with \(PX = XP\) and \(X^H = sX\) constraints. (English) Zbl 1221.15025

The matrix equation \(AX=B\) with \(PX=XP\) and \(X^H=sX\) is considered, where \(P\) is a given Hermitian involutory matrix and \(s=+1\).
It is shown that by an eigenvalue decomposition of \(P\) one obtains a pair of similar problems which is equivalent with the initial one.
Necessary and sufficient conditions for the existence of a solution are provided, as well as a representation of the general solution.
Similar results are derived for eigenvalue-free solutions to the considered constrained problem.
The optimal approximation problem \(\min\|X-E\|_F\) is studied, where \(E\in\mathbb{C}^{n\times n}\) is a given matrix and \(X_s\) is the solution set of the constrained problem.
Eigenvalue-involved and eigenvalue-free solutions for the optimal problem are provided.
A similar approach is presented for the extended problem of the matrix equations \(AX=B,\;XC=D,\;A,B\in\mathbb{C}^{m\times n},\;C,D\in\mathbb{C}^{n\times p}\), with the same constraints.
Some numerical examples illustrate the effectiveness of the proposed methods.

MSC:

15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
65F30 Other matrix algorithms (MSC2010)
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