An eigenvalue problem for derogatory matrices

[摘要] A matrix A is called derogatory if there is more than one Jordan submatrix associated with an eigenvalue lambda. In this paper, we are concerned with the eigenvalue problem of this type of matrices. The singularities of the resolvent of A : R(z) = (A-zl)(-1) are exactly the eigenvalues of A. Let us consider the Laurent series of R expanded at lambda and denote its coefficients c(k) (-infinity <= k <= infinity). D := c(-2) is the nilpotent operator, that is, there exists the order l of lambda such that D-l := c(-l-1) = 0 (l >= 1). Additionally, for an arbitrary vector z, D-z(l-1) is an eigenvector of lambda. Then lambda is computed from the corresponding eigenvector Dl-1 z. In order to estimate the integral representation of D(k)z, we apply the trapezoidal rule on the circle enclosing lambda but excluding other eigenvalues of A. It is our result that, so far as related linear equations are solved with necessary precision, the eigenvalues of derogatory matrices can be computed numerically as exactly as we want and so are corresponding (generalized) eigenvectors, too. (c) 2006 Elsevier B.V. All rights reserved.

[发布日期] 2007-02-15

[效力级别]  Proceedings Paper

[关键词] multiple eigenvalue;Jordan canonical form;generalized eigenvector