Let Pm(z) be a matrix polynomial of degree m whose coefficients At <FONT FACE=Symbol>Î</FONT> 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 <FONT FACE=Symbol>Î</FONT> Cp×q, R <FONT FACE=Symbol>Î</FONT> Cp×n, Z = diag (z1,...,z n) with z i <FONT FACE=Symbol>¹</FONT> z j for i <FONT FACE=Symbol>¹</FONT> j, 0 < |z j| < 1, and L <FONT FACE=Symbol>Î</FONT> 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.
matrix polynomials; block companion matrices; departure from normality; eigenvalue sensitivity; controllability Gramians
