$S_n$矩陣的數值域是一個圓盤，我們想知道第$k$層的數值域是否也是圓盤。我們讓$S_5$矩陣的特徵質屬於實數和數值域為圓盤。
如果$S_n$結合Blaschke product $B$，並且$B$等於$C$合成$D$，其中$C$的degree是2、$D$的degree是3。我們會得到$S_5$的第2層也會是圓，$S_5$的第3層會是單點。
$A$和$B$是2乘2矩陣，我們有$w(A+B)\leq w(A)+w(B)$基本的不等式。我們對在等號成立時感到興趣。 然而我們得到等號成立時，$A$和$B$矩陣必須滿足一些充分必要條件。

For an $S_n$-matrix with a circular disc as its numerical range, we want to know whether its rank-$k$ numerical range is also a circular disc. We show that, for an $S_5$-matrix $A$ with real spectrum and circular numerical range, if its associated Blaschke product $B$ has a normalized decomposition $B=C\circ D$, with $C$ of degree 2 and $D$ of degree 3, then $\Lambda_2(A)$ is also a circular disk and $\Lambda_3(A)$ is singleton (cf. Theorem 3.3). For $A$ and $B$ be $2\times2$ matrices, we have $w(A+B)\le w(A)+w(B)$. We are interested in when it becomes
equality. We obtain a necessary and sufficient condition for $w(A+B)= w(A)+w(B)$ to hold (cf. Proposition 4.3).

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