| Summary: | 碩士 === 中原大學 === 數學研究所 === 86 === The numerical range has been studied extensively in the last few decades.This concept is very useful for studying matrices and operators, and has alot of applications to other subjects. However, we know of no book dedicatedto presenting the fundamentals of this concept. The purpose of this thesisis to arrange a few interesting and useful concepts which were dispersed invariant papers or books. This thesis also contains some new observationswhich will be indicated. We mainly only discuss finite-dimensional results. This thesis consists of the following sections. Section 1 is introduction.In Section 2, we define the numerical range of a matrix and give a fewproperties of numerical range. In Section 3, we define the numerical radiusof a matrix and give a few properties of numerical radius. In Section 4, weobserve that if the numerical range of a matrix A is contained in theinterior of a triangle inscribed in the unit circle, then A is a strictcontraction. In Section 5, we give a few characterization of normal matricesin terms of the numerical range and give a few concepts which concernseigenvalues lying on the boundary of the numerical range. We alsocharacterize matrices for which the numerical range of a matrix and theconvex hull of spectrum coincide. In Section 6, we recall the well-knownGersgorin's Theorem. Such set containment was first obtained by Gersgorinfor the spectrum. This is perhaps quite an important motivation in obtaininga Gersgorin-type set containing the numerical range of a matrix. It is thegoal of this section to present for the numerical range an analogue ofGersgorin's Theorem. In Section 7, we will look at some mapping theorems forthe numerical range W(A) and relate W(f(A)) and W(A) when the function f isanalytic in various regions.
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