在此論文中，我們將探討關於「加權排列矩陣」之數值域邊界有直線的等價條件，以及「加權位移矩陣」之數值域半徑與其weights排列順序之間的關係。首先，我們發現ㄧ個3*3的加權排列矩陣A，它的數值域W(A)會2π/3對稱，也就是說W(A)邊界有一條線，其圖形為一個三角形，若且為若，A是ㄧ個正規(normal)矩陣。如果A是ㄧ個4*4的加權排列矩陣，同樣地，它的數值域W(A)會2π/4對稱，若W(A)邊界有一條線，其圖形將為一個四邊形，若且為若，A可被分解為兩個2*2的加權排列矩陣。除此之外，我們發現一個4*4伴隨矩陣其數值域邊界有四條線的等價條件就是此伴隨矩陣可以被分解成兩個2*2的加權排列矩陣。
另外，我們已知一個n*n加權位移矩陣A的數值域W(A)為一個以原點為圓心的圓盤，我們發現n=4時，其數值域半徑r(A)最大若且為若|a2|為所有weights絕對值中的最大值；n=5時，其數值域半徑r(A)最大之等價條件則是|a2|或是|a3|為所有weights絕對值中的最大值。

In this thesis, we will study about numerical ranges of
weighted permutation matrices and weighted shift matrices. Firstly,
we know that if $A$ is a $3 imes3$ weighted permutation matrix,
$W(A)$ has symmetry of $frac{2pi}{3}$. Thus, if there is a line
segment on $partial W(A)$ then $W(A)$ is a triangle. Moreover, $A$
is normal. If $A$ is a $3 imes3$ weighted permutation matrix,
$W(A)$ has symmetry of $frac{2pi}{4}$. If there is a line segment
on $partial W(A)$ then $W(A)$ is a quadrangle. Moreover, $Acong
A_{1}oplus A_{2}$, where $A_{1}$ and $A_{2}$ are $2 imes 2$
weighted permutation matrices. Let $A$ be a $4 imes4$ companion
matrix. We will see that $W(A)$ has four line segments if and only
if $A$ can be reducible.
Another subject is that we are interested in finding the order of
the weights of a weighted shift matrix so that the numerical radius
will be the largest.

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