在本論文中，我們首先學習一些關於友矩陣數值域的基本性質。參考文獻1特別探討可分解的友矩陣，同時還證明一個友矩陣的數值域是以原點為圓心的圓盤，其充分必要條件在於這個友矩陣是Jordan區塊。而我們在此僅針對那些數值域為橢圓形的可分解友矩陣作討論，並試圖給這些矩陣一個完整的特徵。
從論文第三節可以看出所有的4 × 4可分解友矩陣將完全被解決，原因是我們會證明一個4 × 4可分解友矩陣的數值域是橢圓，若且為若，這個矩陣的光譜為{a,-a,i/a,-i/a}，其中|a|≧sqrt(1+sqrt(2))；或者這個矩陣的光譜為{a,ai,-1/a,-i/a}，其中|a|≧1+sqrt(2)。最後，我們在論文的第四節就把討論的對象擴大為6 × 6可分解友矩陣。

In this thesis, we study some properties of numerical ranges of companion matrices. Previous works [1] in this respect are the criterion for these matrices to be reducible and show that the numerical range of a companion matrix is a circular disc centered at the origin if and only if the matrix equals the Jordan block. Here we want to give a complete characterization for reducible companion matrices with elliptical numerical range.
In Section 3, 4 × 4 reducible companion matrices will be completely solved. We show that a 4 × 4 reducible companion matrix A has an ellipse as its numerical range if and only if either σ(A)＝{a,-a,i/a,-i/a} where |a|≧sqrt(1+sqrt(2)), or σ(A)＝{a,ai,-1/a,-i/a} where |a|≧1+sqrt(2). Here σ(A) denotes the spectrum of the matrix A. In Section 4, we discuss the cases for 6 × 6 reducible companion matrices.

[1] Hwa–Long Gau and Pei Yuan Wu, Companion matrices: reducibility, numerical ranges and similarity to contractions, Linear Algebra Appl., 383 (2004), 127–142.
[2] U.Haagerup, P. de la Harpe, The numerical radius of a nilpotent operator on a Hilbert space, Proc. Amer. Math. Soc. 115 (1992) 371–379.
[3] R. A. Horn and C. R. Johnson. Matrix analysis, Cambridge University Press, Cambridge, 1985.
[4] R. A. Horn and C. R. Johnson. Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.
[5] D. S. Keeler, L. Rodman and I. M. Spitkovsky, The numerical range of 3 × 3 matrices, Linear Algebra Appl., 252 (1997), 115–139.