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**Symmetric matrix polynomial equations.**
*(English)*
Zbl 0599.93008

The following matrix polynomial equation is studied (1) \(A^*X+X^*A=2B\), where A,B,X are real \(n\times n\) matrix polynomials, and \(A^*(s)=A(-s)\). Here A and B are given and X is unknown. It is assumed that A is stable, i.e. det A has no zeros in the closed right half-plane, and that \(XA^{-1}\) is proper. A general form of solutions is given, and particular attention is paid to the case when B(s) is positive definite for pure imaginary s. The discrete analogue of (1) is considered as well (in this case \(A^*(s)=A(s^{-1})\), and instead of being polynomials the matrix functions A,B,X are of type \(\sum^{p}_{j=-p}s^ jA_ j\), where \(A_ j\) are \(n\times n\) matrices). Numerical algorithms for solving (1) and its discrete counterparts are devised and tested.

Reviewer: L.Rodman

### MSC:

93B25 | Algebraic methods |

15A24 | Matrix equations and identities |

65F30 | Other matrix algorithms (MSC2010) |

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

### References:

[1] | V. Kučera: Discrete Linear Control - The Polynomial Equation Approach. Academia, Prague 1979. |

[2] | V. Kučera, M. Šebek: A polynomial solution to regulation and tracking, parts I, II. Kybernetika 20 (1984), 3, 177-188 and 4, 257-282. · Zbl 0542.93013 |

[3] | V. Kučera: Asymptotic LQG control of singular systems. IEEE Trans. Automat. Control · Zbl 0584.93076 |

[4] | Z. Vostrý: New algorithm for polynomial spectral factorization with quadratic convergence, parts I, II. Kybernetika 11 (1975), 6, 415-422 and 12 (1976), 4, 248-259. · Zbl 0333.65023 |

[5] | J. Ježek: Conjugated and symmetric polynomial equations, parts I, II. Kybernetika 19 (1983), 2, 121-130 and 3, 196-211. |

[6] | J. Ježek, V. Kučera: Efficient algorithm for spectral factorization. Preprints of the 9th IFAC World Congress, Budapest 1984, vol. IX, pp. 38-43. |

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