docs: 圆盘定理初步完成，添加部分习题 #72
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yhwu-is
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Sep 2, 2024
讲义/专题/15 相似标准形：动机与基础.tex
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| @@ -843,6 +843,96 @@ \section{实数域与复数域的讨论} | |||
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| \section{特征值的估计} | |||
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| 对于低阶的矩阵来说，我们可以通过解特征多项式来精确求得特征值；但对于高阶矩阵而言，解特征多项式是非常困难的，所幸相关的工作一般来说也不需要我们精确求得特征值，所以我们可以通过一些方法来估计特征值. | |||
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| \begin{definition}{}{Gershgorin disks} | |||
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这个是 label，定理的中文名字我也加一下吧
yhwu-is
reviewed
Sep 2, 2024
讲义/专题/15 相似标准形：动机与基础.tex
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| 设 $T \in \mathcal{L}(V)$，并且 $v_1, \ldots, v_n$ 是 $V$ 的一组基，$D_i, i = 1, \ldots, n$ 是 $T$ 关于这组基的 Gershgorin 圆盘. 若 $\cup_{i = 1}^n D_i$ 是 $k$ 个不相交的连通区域 $R_1, \ldots, R_k$ 的并，并且 $R_r$ 是一个 $m_r$ 个 Gershgorin 圆盘的并，那么 $T$ 有 $m_r$ 个特征值落在 $R_r$ 中，$r = 1, \ldots, k$. | ||
| \end{theorem} | ||
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| 这一定理的证明需要用到特征值的连续性，这里不再赘述. 借助于这一条定理，我们限制了每个连通区域内特征值的个数，从而使得我们可以更好地估计特征值的位置. 而自然地，我们也可以利用 Gershgorin 圆盘来刻画一些与特征值有关的性质. |
45gfg9
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Sep 3, 2024
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