本論文探討一個四階方陣A，其高秩數值域的幾何圖形是什麼樣的圖形。我們將四階方陣的秩二數值域分類。對於一個四階方陣A，我們經由考慮A的associated polynomial來對秩二數值域作分類。對於每一個分類，我們將完整地描述它們的幾何圖形。 Let $A$ be an $n$-by-$n$ matrix. For $1leq k leq n$, the rank-$k$ numerical range of $A$ is defined and denoted by $Lambda_k(A) = {lambdainmathbb{C}: PAP=lambda P mbox{ for some rank-{it k} orthogonal projection $P$}}$. In this thesis, we give a complete description of the higher-rank numerical ranges of $4$-by-$4$ matrices. We classify the rank-$2$ numerical ranges of $4$-by-$4$ matrices. Our classification is based on the factorability of the associated polynomial $p_A(x,y,z)equiv mathrm{det}(xmathrm{Re,}A + ymathrm{Im,}A + zI_4)$ of a $4$-by-$4$ matrix $A$. For each class, we also completely determine the shape of the rank-$2$ numerical range of a $4$-by-$4$ matrix.
