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**On the parametric solution to the second-order Sylvester matrix equation \(EVF^{2} - AVF - CV=BW\).**
*(English)*
Zbl 1160.15017

For a real diagonalizable matrix \(F\), a controllable matrix triple \((E,A,B)\), and a given \(C\), the authors describe a parametric solution \(V,W\) for the second order Sylvester equation

\[ EVF^2 - AVF - CV = BW. \]

If \(F\) is not explicitly given, the solution is found by elementary transformations performed on respective matrix pencils and the solution depends on the eigenvalues of \(F\), as well as on another set of parameters. If the eigenvalues of \(F\) are prescribed, then the solution method involves the singular value decomposition.

\[ EVF^2 - AVF - CV = BW. \]

If \(F\) is not explicitly given, the solution is found by elementary transformations performed on respective matrix pencils and the solution depends on the eigenvalues of \(F\), as well as on another set of parameters. If the eigenvalues of \(F\) are prescribed, then the solution method involves the singular value decomposition.

Reviewer: Frank Uhlig (Auburn)

### MSC:

15A24 | Matrix equations and identities |

65F30 | Other matrix algorithms (MSC2010) |

93B40 | Computational methods in systems theory (MSC2010) |

93B05 | Controllability |

### Keywords:

controllability; second-order Sylvester matrix equation; parametric solution; singular value decomposition
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\textit{G.-S. Wang} et al., Math. Probl. Eng. 2007, Article ID 21078, 16 p. (2007; Zbl 1160.15017)

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### References:

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