Eigenvector-free solutions to the matrix equation AXBH=E with two special constraints

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. © 2013 yuyang qiu.