Two fundamental concepts of geometric control theory, (A,B) -invariant and controllability subspaces, are discussed in terms of spaces spanned by closed-loop eigenvectors. Included is a characterization of V^{\ast},R\Re^{\ast} , the supremal (A,B) -invariant and controllability subspaces contained in the kernel of some map. Applying ideas found in numerical analysis literature, it is shown that, for design purposes, knowledge of V^{\ast},R\Re^{\ast} is not sufficient: certain subspaces of V^{\ast},R\Re^{\ast} may be useless with respect to true design applications. Possible consequences of design based on these unreliable parts of V^{\ast},R\Re^{\ast} are discussed. Finally, prototype algorithms for computing basis vectors for V^{\ast},R\Re^{\ast} are given. Their strength is in the additional information which makes it possible to identify the reliable components of V^{\ast},R\Re^{\ast} Numerical stability and efficiency are "built in" to the algorithms through the use of routines which have been implemented, tested thoroughly, and recommended by recognized experts in numerical analysis.

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