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Computational & Applied Mathematics

versão On-line ISSN 1807-0302

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BAZAN, Fermin S. Viloche. Matrix polynomials with partially prescribed eigenstructure: eigenvalue sensitivity and condition estimation. Comput. Appl. Math. [online]. 2005, vol.24, n.3, pp. 365-392. ISSN 1807-0302.

Let Pm(z) be a matrix polynomial of degree m whose coefficients At Î Cq×q satisfy a recurrence relation of the form: hkA0+ hk+1A1+...+ hk+m-1Am-1 = hk+m, k > 0, where hk = RZkL Î Cp×q, R Î Cp×n, Z = diag (z1,...,zn) with zi ¹ zj for i ¹ j, 0 < |zj| < 1, and L Î Cn×q. The coefficients are not uniquely determined from the recurrence relation but the polynomials are always guaranteed to have n fixed eigenpairs, {zj,lj}, where lj is the jth column of L*. In this paper, we show that the zj's are also the n eigenvalues of an n×n matrix CA; based on this result the sensitivity of the zj's is investigated and bounds for their condition numbers are provided. The main result is that the zj's become relatively insensitive to perturbations in CA provided that the polynomial degree is large enough, the number n is small, and the eigenvalues are close to the unit circle but not extremely close to each other. Numerical results corresponding to a matrix polynomial arising from an application in system theory show that low sensitivity is possible even if the spectrum presents clustered eigenvalues.

Palavras-chave : matrix polynomials; block companion matrices; departure from normality; eigenvalue sensitivity; controllability Gramians.

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