在這篇論文中，我們主要討論具備怎樣性質的n×n矩陣A與m×m矩陣B能讓這個等式"w(A\otimes B)=" ‖A‖w(B)成立，其中w(∙)及‖∙‖分別代表一個矩陣的數值半徑(numerical radius)及範數(norm)。我們證明了以下結果：(1)假如A是一個S_n矩陣，則"w(A\otimesB)=" w(B)的充分必要條件是B的數值域(numerical range)是個圓心在原點的圓盤並且k_B≤n，其中k_B這個參數指的是在B的壓縮矩陣中數值域與B相同，這種壓縮矩陣尺寸的最小值；以及(2)若A是個範數為1的completely nonunitary矩陣，而m×m矩陣B滿足k_B=m，則"w(A\otimes B)=" w(B)的充分必要條件是B的數值域是個圓心在原點的圓盤並且k_B≤p_A+1，其中p_A這個參數指的是讓‖A^k ‖=‖A‖^k成立，所有k的最大值。在上述的情況下，我們都得到"A\otimes B" 的數值域與B的數值域相同。接下來，我們也對友矩陣(companion matrix)作一些討論，我們證明：若A是一個n×n的友矩陣，則"W(A\otimes A)" 是個圓心在原點的圓盤的充分必要條件是A是一個n×n的Jordan block J_n.

In this thesis, we characterize matrices A in M_n and B in M_m which yield the equality w(A\otimes B)=\|A\|w(B), where w(
.) and \|.\| denote, respectively, the numerical radius and the operator norm of a matrix. We show that (1) if A is an Sn-matrix, then w(A\otimes B)=w(B) if and only if the numerical range W(B) of B is a circular disc
centered at the origin and k_B\leq n, where
k_B=min{k:W(V*BV)=W(B) for some V in M_mk with V*V=I_k};
and (2) if A is completely nonunitary with \|A\|=1 and k_B =m, then w(A\otimes B)=w(B) if and only if W(B) is a circular disc centered at the origin and k_B\leq pA+1,
where p_A=sup{k:\|A\|^k=\|A^k\|}
In the above cases, we all have W(A\otimes B)=W(B). Next, we consider the class
of companion matrices. We prove that if A is an n-by-n companion matrix, then
W(A\otimes A) is a circular disc centered at the origin if and only if A is equal to the
n-by-n Jordan block J_n.

[1] M. D. Choi, C.K. Li, Constrained unitary dilations and numerical ranges, J.Operator Theory 46 (2001), pp. 435-447.
[2] H.-L. Gau, Numerical ranges of reducible companion matrices, Linear Algebra Appl. 432 (2010), pp. 1310-1321.
[3] H.-L. Gau and P. Y. Wu, Numerical range of S(\phi), Linear Multilinear Algebra 45 (1998), pp. 49-73.
[4] H.-L. Gau, P. Y. Wu., Lucas' theorem refined, Linear Multilinear Algebra, 45 (1999), pp. 359-373.
[5] H.-L. Gau and P. Y. Wu, Condition for the numerical range to contain an elliptic disc, Linear Algebra Appl. 364 (2003), pp. 213-222.
[6] H.-L. Gau and P. Y. Wu, Finite Blaschke products of contractions, Linear Algebra Appl. 368 (2003), pp. 359-370.
[7] H.-L. Gau, P. Y. Wu, Numerical range circumscribed by two polygons, Linear Algebra Appl. 382 (2004), pp. 155-170.
[8] H.-L. Gau and P. Y. Wu, Companion matrices: reducibility, numerical ranges and similarity to contractions, Linear Algebra Appl. 383 (2004), pp. 127-142.
[9] H.-L. Gau and P. Y. Wu, Numerical ranges of companion matrices, Linear Algebra Appl. 421 (2007), pp. 202-218.
[10] H.-L. Gau and P. Y. Wu, Structures and numerical ranges of power partial isometries, Linear Algebra Appl. 440 (2014), pp. 325-341.
[11] H.-L. Gau, C.K. Li, and P. Y. Wu, Higher-rank numerical ranges and dilations, J. Operator Theory 63:1 (2010), pp. 181-189.
[12] H.-L. Gau, K.-Z. Wang and P. Y. Wu., Numerical radii for tensor products of operators, Integral Equations and Operator Theorey, 78 (2014), pp. 375-382.
[13] H.-L. Gau, K.-Z. Wang and P. Y. Wu., Numerical radii for tensor products of matrices, Linear Multilinear Algebra, (2014), to appear.
http://dx.doi.org/10.1080/03081087.2013.839669
[14] K. Gustafson and D. K. M. Rao, Numerical Range. The Field of Values of Linear
Operators and Matrices, Springer, New York, 1997.
[15] P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982.
[16] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University
Press, Cambridge, 1991.
[17] R. Kippenhahn, Uber den Wertevorrat einer Matrix, Math. Nachr., 6 (1951), pp. 193-228.
[18] P. Y. Wu, Unitary dilations and numerical ranges, J. Operator Theory 38 (1997), pp. 25-42.
[19] P. Y. Wu, Numerical ranges as circular discs, Applied Math. Lett. 24 (2011), pp. 2115-2117.
[20] P. Y. Wu, H.-L. Gau and M.-C. Tsai, Numerical radius inequality for C_0 contractions, Linear Algebra Appl. 430 (2009), pp. 1509-1516.