We study the problem of computing the eigenstructure of a polynomial matrix. Via a backward error analysis we analyze the stability of some block Toeplitz algorithms to obtain this eigenstructure. We also elaborate on the nature of the problem, i.e. conditioning and posedness.

Abstract-A simple necessary and sufficient algebraic condition is given to detect zeros at infinity in a polynomial matrix.

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The problem of decoupling and complete pole assignment of linear square, and controllable systems by static state feedback is addressed in this paper. Based on a characterization of the whole set of attainable finite pole-zero structures of a decouplable system, we present a reliable numerical algorithm which tests the conditions for decoupling and computes the state feedback which decouples the system with a particular pole-zero finite structure, avoiding unnecessary cancellations of invariant zeros. With the use of this algorithm, fixed decoupling poles are determined, non-fixed poles can be arbitrarily located, and no cancellation of system invariant zeros is produced, if this is not necessary for decoupling.