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<title>关于聚集特征值的评述</title>
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<b>下一节：</b><a name="tex2html3041" href="node154.html">关于高相对精度特征值计算的评述</a>
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关于聚集特征值的评述</a>
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当特征值 <span class="math-inline">\lambda</span> 与矩阵 <span class="math-inline">A</span> 的一个或多个其他特征值非常接近时，换句话说，当 <span class="math-inline">\lambda</span> 属于聚类特征值时，根据（<a href="node151.html#ineq:err-1-ev">4.54</a>）的保证，只要 <span class="math-inline">\Vert r\Vert _2</span> 很小，计算得到的 <span class="math-inline">\widetilde\lambda</span> 仍然是准确的，但由于（<a href="node152.html#ineq:err-1-evt">4.56</a>）中分母出现的间隙 <span class="math-inline">\delta</span>，计算得到的特征向量 <span class="math-inline">\widetilde x</span> 可能不准确。事实证明，与聚类特征值相关的每个单独特征向量对扰动非常敏感，但由所有这些特征向量张成的特征空间则不然。因此，对于聚类特征值，我们应该计算整个特征空间。
基于上述思路的理论可以建立，从残差矩阵开始
<div class="math-display">R=A\widetilde X-\widetilde X\widetilde\Lambda,</div>
其中 <span class="math-inline">\widetilde\Lambda</span> 是对角矩阵，其对角元素是聚类中所有特征值的近似值，而 <span class="math-inline">\widetilde X</span> 的列是对应的近似特征向量。假设 <span class="math-inline">\widetilde X</span> 的列是正交的，并且 <span class="math-inline">\widetilde\Lambda</span> 比 <span class="math-inline">\Lambda</span> 更接近，<span class="math-inline">\Lambda</span> 是对角矩阵，其对角元素是聚类中的所有特征值。设 <span class="math-inline">X</span> 是与 <span class="math-inline">\Lambda</span> 相关的特征向量矩阵，<span class="math-inline">\delta</span> 是 <span class="math-inline">\widetilde\Lambda</span> 对角线上任意近似特征值与 <span class="math-inline">A</span> 中未出现在 <span class="math-inline">\Lambda</span> 对角线上的特征值之间的最小差值。那么 [<a href="node421.html#daka:70">101</a>]
<div class="math-display">\Vert\sin\Theta(X,\widetilde X)\Vert _{F}\le\frac {\Vert R\Vert _{F}}{\delta},</div>
其中 <span class="math-inline">\Theta(X,\widetilde X)</span> 是对角矩阵，其对角元素是 <span class="math-inline">X^*\widetilde X</span> 的奇异值的反余弦值。由于其定义方式，预计这个间隙会很大，因此只要 <span class="math-inline">\Vert R\Vert _{F}</span> 很小，<span class="math-inline">\Vert\sin\Theta(X,\widetilde X)\Vert _{F}</span> 也会很小。
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<b>上一级：</b><a name="tex2html3035" href="node148.html">稳定性和准确性评估 </a>
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<address>
Susan Blackford
2000-11-20
</address>
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