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Matrix equation $$AXB=E$$ with $$PX=sXP$$ constraint. (English) Zbl 1150.15008
Summary: The matrix equation $$AXB=E$$ with the constraint $$PX=sXP$$ is considered, where $$P$$ is a given Hermitian matrix satisfying $$P^2=I$$ and $$s=\pm 1$$. By an eigenvalue decomposition of $$P$$, the constrained problem can be equivalently transformed to a well-known unconstrained problem of a matrix equation whose coefficient matrices contain the corresponding eigenvector, and hence the constrained problem can be solved in terms of the eigenvectors of $$P$$. A simple and eigenvector-free formula of the general solutions to the constrained problem by generalized inverses of the coefficient matrices $$A$$ and $$B$$ is presented. Moreover, a similar problem of the matrix equation with generalized constraint is discussed.

MSC:
 15A24 Matrix equations and identities 15A18 Eigenvalues, singular values, and eigenvectors 15A09 Theory of matrix inversion and generalized inverses
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References:
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