Perturbations of Jordan matrices

Abstract

We consider perturbations of a large Jordan matrix, either random and small in norm or of small rank. In the case of random perturbations we obtain explicit estimates which show that as the size of the matrix increases, most of the eigenvalues of the perturbed matrix converge to a certain circle with centre at the origin. In the case of finite rank perturbations we completely determine the spectral asymptotics as the size of the matrix increases.

Spectral properties of matrices associated with some directed graphs

“…The results that we attain are to some extent analogous to calculations in [1,4,5,7,9] but the details are different. We will often need to count the number of zeros that an analytic function has in a particular region.…”

Large bidiagonal matrices and random perturbations

“…M. Hager and E.B. Davies [6] considered the case of large Jordan block matrices subject to small Gaussian random perturbations and showed that with a sufficiently small coupling constant most eigenvalues can be found near a circle, with probability close to 1, as the dimension of the matrix N gets large. Furthermore, they give a probabilistic upper bound of order log N for the number of eigenvalues in the interior of a circle.…”

Long transient oscillations in a class of cooperative cellular neural networks

“…It is crucial that c 1 , c 2 , and Q are independent of M and also of the cell time constant τ . Proof The proof depends on finer properties of the auxiliary equation and the auxiliary matrix associated. The main ingredients are exploiting the logarithmic singularity of the auxiliary equation and referring to the root asymptotics results on certain parameterized families (of products) of lacunary polynomials. The essential observation is that the auxiliary matrix defined by the type one central wave is a rank two perturbation of a 2 M × 2 M matrix of the form , where stays for the translation time corresponding to the normalized time constant τ = 1 s in the CNN model (1) and J is the usual Jordan matrix composed of 0's everywhere except for the superdiagonal which is composed of 1's.…”