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The reflexive and anti-reflexive solutions of the matrix equation $$AX=B$$. (English) Zbl 1050.15016
Let $$P\in\mathbb{C}^{n\times n}$$ be both Hermitian and unitary. A matrix $$X\in\mathbb{C}^{n\times n}$$ is said to be reflexive (resp. anti-reflexive) with respect to $$P$$ when $$X= PXP$$ (resp. $$X = -PXP$$). Conditions are given for existence of reflexive and antireflexive solutions (with respect to $$P$$) of the equation (1) $$AX= B$$, where $$A,B\in\mathbb{C}^{m\times n}$$ are given matrices. If $$X_0\in \mathbb{C}^{n\times n}$$ is a given matrix, expressions are given for the reflexive and anti-reflexive solutions of (1) which are the nearest to $$X_0$$ in the Frobenius norm.

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