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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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