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 sucient, 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 sucient 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.

[1] C.-H. Chang, H.-L. Gau, K.-Z. Wang, Equality of higher-rank numerical ranges
of matrices, Linear Multilinear Algebra, 62 (2014), 626{638.
[2] W.-S. Cheung, C.-K. Li, Elementary proofs for some results on the circular
symmetry of the numerical range, Linear Algebra Appl., 61 (2013), 596{602.
[3] M.-T. Chien, B.-S. Tam, Circularity of the numerical nange, Linear Algebra
Appl., 201 (1994), 113{133.
[4] H.-L. Gau, K.-Z. Wang, P. Y. Wu, Numerical radii for tensor prod-
ucts of matrices, Linear Multilinear Algebra, (2014), to appear.
http://dx.doi.org/10.1080/03081087.2013.839669
[5] K. E. Gustafson, D. K. M. Rao, Numerical Range, the the Field of Values of
Linear Operators and Matrices, Springer, New York, 1997.
[6] P. R. Halmos, A Hilbert Space Problem Book, Springer, New York, 1982.
[7] D. S. Keeler, L. Rodman, I. M. Spitkovsl, The numerical range of 3 x 3 matrices,
Linear Algebra Appl., 252 (1997), 115{139.
[8] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis. Cambridge, Cambrige
University Press, 1991.
[9] P. Y. Wu, Numeaical ranges as circular discs, Appl. Math. Lett., 24 (2011), 2115{
2117.
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