## 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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### References:

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