K-spectral Sets and Functions of Nonnormal Matrices

Abstract

In this thesis, we study K-spectral sets and use them to bound norms of functions of nonnormal matrices. For a fixed constant K > 0, the set Ω is said to be a K-spectral set for a matrix A if the spectrum Λ(A) is contained in Ω and the inequality ∥f(A)∥ ≤ K supz∈Ω |f(z)| holds for all rational functions defined on Ω with poles outside of Ω. K-spectral sets are useful for studying nonnormal matrices, for which the asymptotic behavior of ∥f(A)∥ suggested by the eigenvalues may not agree well with the short-time or transient behavior. We extend a result of Crouzeix and Palencia [17], who show that the numerical range W(A) = {⟨v∗Av⟩ : v ∈ Cn,∥v∥ = 1} is a (1+√2)-spectral set for any n×n matrix A, to sets in the complex plane which do not necessarily contain W(A). This allows us to study more general K-spectral sets of interest, such as disks or half-planes. We also find some special cases in which the constant (1+√2) for W(A) can be replaced by 2, which is the value conjectured by Crouzeix. Additionally, we also provide details of our numerical studies related to Crouzeix’s conjecture and K-spectral sets. This includes the construction of functions using numerical conformal mapping and optimization procedures to find bounds for K-spectral sets, as well as other bounds found via our extension of the Crouzeix-Palencia arguments or the numerical construction of near-normal dilations of matrices. We compare different analytical and numerical bounds found using these methods, and illustrate how these can be used for potential applications of interest.