Eigenvalue Estimates via Pseudospectra

Generalizing Eigenvalue Theorems to Pseudospectra Theorems

Structured Pseudospectra for Polynomial Eigenvalue Problems, with Applications

Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structure...

Backward errors and pseudospectra for structured nonlinear eigenvalue problems

Minimal structured perturbations are constructed such that an approximate eigenpair of a nonlinear eigenvalue problem in homogeneous form is an exact eigenpair of an appropriately perturbed nonlinear matrix function. Structured and unstructured backward errors are compared. These results extend previous results for (structured) matrix polynomials to more general functions. Structured and unstru...

Eigenvalue condition numbers and Pseudospectra of Fiedler matrices

The aim of the present paper is to analyze the behavior of Fiedler companion matrices in the polynomial root-finding problem from the point of view of conditioning of eigenvalues. More precisely, we compare: (a) the condition number of a given root λ of a monic polynomial p(z) with the condition number of λ as an eigenvalue of any Fiedler matrix of p(z), (b) the condition number of λ as an eige...

Structured Pseudospectra and the Condition of a Nonderogatory Eigenvalue

Let λ be a nonderogatory eigenvalue of A ∈ C. The sensitivity of λ with respect to matrix perturbations A A + ∆,∆ ∈ ∆, is measured by the structured condition number κ∆(A,λ). Here ∆ denotes the set of admissible perturbations. However, if ∆ is not a vector space over C then κ∆(A, λ) provides only incomplete information about the mobility of λ under small perturbations from ∆. The full informati...