On The Computation Of Eigenvalue Assignment Problem

This note shows that in computing the feedback gain for eigenvalue assignment in multivariable systems, the reduction of the condition number of the closed loop eigenvector matrix (CLEM) is a necessary step to achieve some highly important and desirable technical and computational properties. This note also shows that the reduction of the condition number of other relevant matrix does not sufficiently imply the achievement of these properties. The only existing algorithm capable of reducing the condition number of CLEM computes the explicit CLEM. Fortunately, this computation is very efficient and numerically stable, and is completely different from the computation of the eigenvector matrix of a given matrix.