A note on the inversion of Sylvester matrices in control systems. (English) Zbl 1217.65058

Summary: We give a sufficient condition (the solvability of two standard equations) of a Sylvester matrix by using the displacement structure of the Sylvester matrix, and, according to the sufficient condition, we derive a new fast algorithm for the inversion of a Sylvester matrix, which can be denoted as a sum of products of two triangular Toeplitz matrices. The stability of the inversion formula for a Sylvester matrix is also considered. The Sylvester matrix is numerically forward stable if it is nonsingular and well conditioned.


65F05 Direct numerical methods for linear systems and matrix inversion
65K10 Numerical optimization and variational techniques
93C05 Linear systems in control theory
15B05 Toeplitz, Cauchy, and related matrices
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