在這篇論文中，我們去考慮Jm與矩陣A的張量積的數值域半徑和矩陣A的數值域半徑之間的關係，其中Jm是一個m乘m的喬登方塊。針對m等於2和3，對於Jm與矩陣A的張量積的數值域半徑等於矩陣A的數值域半徑時，得到不同的充分必要條件。我們證明J2與矩陣A的張量積的數值域半徑等於矩陣A的數值域半徑的充分必要條件是矩陣A有一個2乘2的壓縮矩陣B使得B與A的數值域相同且A的數值域是一個以圓點為圓心的圓盤。而且，我們也去證明J3與矩陣A的張量積的數值域半徑等於矩陣A的數值域半徑的充分必要條件是矩陣A有一個3乘3的壓縮矩陣B使得B與A的數值域相同且矩陣A 的數值域是一個以圓點為圓心的圓盤。接下來，保證矩陣A 的數值域是一個圓盤，特別去考慮kA等於2與3時充分必要的關係，其中A經過無數個正交基底變換得到不同大小的矩陣，找到最小的矩陣B使得B與A的數值域相同，這個最小矩陣的大小，定義為kA。若矩陣A 是aij所組成的4 乘4 矩陣，其中aij代表第ｉ列第ｊ行位置上的元素，則上述的這些條件會適用於矩陣Ａ。;In this thesis, we consider the relations between the numerical radius of Jm ⊗ A and A, where Jm is the m-by-m Jordan block.We obtain various conditions, necessary or su cient, for w(Jm ⊗ A) = w(A) to hold for m = 2; 3. We show that w(J2 ⊗ A) = w(A) if and only if A has a 2-by-2 com- pression B such that W(B) = W(A) and W(A) is a circular disc centered at the origin. Moreover, we also show that w(J3 ⊗ A) = w(A) if and only if A has a 3-by-3 compression B such that W(B) = W(A) and W(A) is a circular disc centered at the origin. Next, assume that W(A) is a circular disc centered at the origin, we give the necessary and su cient conditions for kA = 2 and kA = 3, respectively, where kA = min{k ≥ 1 : A has a k × k compression B such that W(B) = W(A)}. Moreover,if A = [aij], i,j = 1,2,3,4, those conditions will be given in terms of aij ′s.
