令D(a, r)代表複數平面上以a為圓心r為半徑的閉圓盤。對任意n乘n的複數矩陣A，其數值域W(A)= {‹Ax, x› : x in C^n, ||x||=1}為複數平面之子集。本計畫之目的在於決定是否存在一n乘n矩陣A使得W(A)為圓盤且W(A)=W(A^2)=…=W(A^n)。我們將先證明在這情況下，圓心必為原點且半徑大於等於1。特別的，我們將對滿足等式W(A)=W(A^(n-1))=D(0, 1)的矩陣A做一完整的刻畫。除此之外，我們也將對滿足等式w(A)=w(A^(n-1))=1的nilpotent矩陣A做一完整的刻畫，此處之w(.)代表數值半徑。 ;Let D(a, r) denote the closed circular disc centered at a with radius r>0 on the complex plane C. For any n-by-n complex matrix A, its numerical range is the subset W(A)={‹Ax, x› : x in C^n, ||x||=1} of the complex plane C. In this project, we want to determine whether there exists an n-by-n matrix A such that W(A)=W(A^2)=…=W(A^n)=D(a,r) for some complex number a and some r>0. In this case, we want to show that the center a must be the origin and the radius r is greater than or equal to 1. In particular, we want to give a complete characterization of A for which the equality W(A)=W(A^(n-1))=D(0, 1) holds. Among others things, if A^n=0, we also want to give a complete characterization of A for which the equality w(A)=w(A^(n-1))=1 holds, where w(.) denotes the numerical radius.
