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<title>正定 B</title>
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正定矩阵 <span class="math-inline">B</span></a>
</h2>
我们将其转化为一个等价的标准厄米特征值问题(HEP)。具体步骤如下：选择 <span class="math-inline">B</span> 的一个分解：
<div class="math-display" id="eq:B_GG">B=GG^*. \tag{5.26}</div>
于是，<span class="math-inline">A - \lambda B</span> 的广义特征值问题等价于 <span class="math-inline">G^{-1}AG^{-*}</span> 的标准HEP。两者共享相同的特征值，因为
<div class="math-display">Ax=\lambda B x\quad\Leftrightarrow\quad G^{-1}AG^{-*} (G^* x)=\lambda (G^* x),</div>
这也意味着，如果 <span class="math-inline">x</span> 是这对矩阵的特征向量，<span class="math-inline">G^* x</span> 就是矩阵 <span class="math-inline">G^{-1}AG^{-*}</span> 的特征向量；反之，如果 <span class="math-inline">y</span> 是 <span class="math-inline">G^{-1}AG^{-*}</span> 的特征向量，<span class="math-inline">G^{-*}y</span> 就是这对矩阵的特征向量。
<span class="math-inline">G</span> 的常见选择包括：
<ol>
<li><span class="math-inline">G=B^{1/2}</span>，即 <span class="math-inline">B</span> 的唯一正定平方根。在这种情况下，<span class="math-inline">G^*=G</span>。这种选择在理论研究中足够好。
</li>
<li><span class="math-inline">G</span> 是乔列斯基(Cholesky)因子；可选地带有枢轴变换，即 <span class="math-inline">G</span> 是下三角矩阵且对角线元素为正。这种选择在数值计算中更受欢迎。
</li>
<li>类似地，<span class="math-inline">G</span> 是上三角矩阵且对角线元素为正。它与第二种选择具有相同的优点。
</li>
</ol>
接下来，有时使用由正定矩阵 <span class="math-inline">M</span> 诱导的内积 <span class="math-inline">(\cdot\,,\,\cdot)_M</span>、相应的向量范数 <span class="math-inline">\Vert\cdot\Vert _M</span> 以及两向量间角度函数（更准确地说，是两个由向量张成的子空间之间的角度）<span class="math-inline">\theta_M(\,\cdot\,,\,\cdot\,)</span> 更为方便。在我们的情况下，<span class="math-inline">M=B</span> 或 <span class="math-inline">B^{-1}</span>。它们的定义如下：
<div class="math-display">
\begin{aligned}
(x,y)_M &\equiv y^*Mx,\\
\Vert x\Vert _M &\equiv \sqrt{(x,x)_M} \equiv \sqrt{x^*Mx},\\
\cos\theta_M(x,y) &\equiv \frac{(x,y)_M}{\Vert x\Vert _M\Vert y\Vert _M}.
\end{aligned}
</div>
当 <span class="math-inline">M=I</span> 时，这三者都简化为通常的定义。不难看出
<div class="math-display" id="ineq:M-2-nrm">\Vert M^{-1}\Vert _2^{-1/2} \Vert x\Vert _2\le\Vert x\Vert _M\le\Vert M\Vert _2^{1/2} \Vert x\Vert _2. \tag{5.27}</div>
通过一些额外的工作，我们可以将 <span class="math-inline">\theta_M</span> 与通常的角度函数联系起来，例如，当 <span class="math-inline">M=I</span> 时，如下所示：
<div class="math-display" id="ineq:aglM-agl">(2\kappa(M))^{-1/2} \sin\theta(x,y)\le\sin\theta_M(x,y)\le (2\kappa(M))^{1/2}\sin\theta(x,y). \tag{5.28}</div>
<p>
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<li><a name="tex2html3443" href="node179.html">残差向量</a>
<li><a name="tex2html3444" href="node180.html">将残差误差转化为向后误差</a>
<li><a name="tex2html3445" href="node181.html">计算特征值的误差界限</a>
<li><a name="tex2html3446" href="node182.html">计算特征向量的误差界限</a>
<li><a name="tex2html3447" href="node183.html">关于聚类特征值的备注</a>
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<b>上一级：</b><a name="tex2html3436" href="node177.html">稳定性与准确性评估</a>
<b>上一节：</b><a name="tex2html3430" href="node177.html">稳定性与准确性评估</a>
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<address>
Susan Blackford
2000-11-20
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