本論文探討一個四階方陣A，其數值域(Numerical Range)邊界上之線段的情形。對A的數值域而言，若PA是可分解的，則其邊界上線段的情形不難解決。所以我們針對PA 是不可分解的情況來推導其邊界上具有一個線段的充分必要條件。另外，具有一對平行邊的情況亦獲得解決，值得注意的是在這個情況下，數值域的邊界最多只有這兩個線段。 In this thesis, we consider the problem of characterizing the numerical ranges of 4 by 4 matrices which have line segments on its boundary. If pA is reducible, we will obtain the result what we want easily. Thus it is sufficient to consider the irreducibility of pA. In this case, we will give a sufficient and necessary condition for the numerical range of a 4 by 4 matrix with a line segment on its boundary. Moreover, we also give a criterion for the numerical range of a 4 by 4 matrix with a pair of parallel line segment on its boundary.