  Galerkin and Petrov-Galerkin methods are some of the most successful solution
procedures in numerical analysis. Their popularity is mainly due to the
optimality properties of their approximate solution. We show that these
features carry over to the (Petrov-)Galerkin methods applied for the solution
of linear matrix equations. Some novel considerations about the use of Galerkin
and Petrov-Galerkin schemes in the numerical treatment of general linear matrix
equations are expounded and the use of constrained minimization techniques in
the Petrov-Galerkin framework is proposed.
