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The matrix equation AXBH=E with SX=XR or PX=sXQ constraint is considered, where S, R are Hermitian idempotent, P, Q are Hermitian involutory, and s=±1. By the eigenvalue decompositions of S, R, the equation AXBH=E with SX=XR constraint is equivalently transformed to an unconstrained problem whose coefficient matrices contain the corresponding eigenvectors, with which the constrained solutions are constructed. The involved eigenvectors are released by Moore-Penrose generalized inverses, and the eigenvector-free formulas of the general solutions are presented. By choosing suitable matrices S, R, we also present the eigenvector-free formulas of the general solutions to the matrix equation AXBH=E with PX=sXQ constraint

journal of applied mathematics

Eigenvector-Free Solutions to the Matrix Equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>A</mml:mi><mml:mi>X</mml:mi><mml:msup><mml:mi>B</mml:mi><mml:mi>H</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:math>with Two Special Constraints

Eigenvector-Free Solutions to the Matrix EquationAXBH=Ewith Two Special Constraints