##
**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.

### MSC:

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 |

PDF
BibTeX
XML
Cite

\textit{H. Li} and \textit{R. Li}, Math. Probl. Eng. 2011, Article ID 609863, 10 p. (2011; Zbl 1217.65058)

### References:

[1] | J. Yang, Z. Xu, and Q. Lu, “A fast algorithm for the inverse of Sylvester matrices,” Journal on Numerical Methods and Computer Applications, vol. 31, no. 2, pp. 92-98, 2010. · Zbl 1240.65088 |

[2] | G. H. Golub and C. F. Van Loan, Matrix Computations, vol. 3 of Johns Hopkins Series in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, Md, USA, 2nd edition, 1989. · Zbl 0733.65016 |

[3] | J. R. Bunch, “The weak and strong stability of algorithms in numerical linear algebra,” Linear Algebra and Its Applications, vol. 88/89, pp. 49-66, 1987. · Zbl 0652.65032 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.