Matrix polynomials with partially prescribed eigenstructure: eigenvalue sensitivity and condition estimation
AUTOR(ES)
Bazán, Fermin S. Viloche
FONTE
Computational & Applied Mathematics
DATA DE PUBLICAÇÃO
2005-12
RESUMO
Let Pm(z) be a matrix polynomial of degree m whose coefficients At Î Cq×q satisfy a recurrence relation of the form: h kA0+ h k+1A1+...+ h k+m-1Am-1 = h k+m, k > 0, where h k = RZkL Î Cp×q, R Î Cp×n, Z = diag (z1,...,z n) with z i ¹ z j for i ¹ j, 0 < |z j| < 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, {z j,l j}, where l j is the jth column of L*. In this paper, we show that the z j's are also the n eigenvalues of an n×n matrix C A; based on this result the sensitivity of the z j's is investigated and bounds for their condition numbers are provided. The main result is that the z j's become relatively insensitive to perturbations in C A 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.
