Have a Question?

施密特正交化 (Schmidt Orthogonalization)

You are here:

1 定义

施密特正交化(Schmidt orthogonalization)是求欧氏空间正交基的一种方法。从欧氏空间(\mathbb{R}^n)任意线性无关的向量组 \mathbf{v}_1,\mathbf{v}_2,\cdots,\mathbf{v}_{m} 出发,求得正交向量组 \mathbf{u}_1,\mathbf{u}_2,\cdots,\mathbf{u}_{m},使由 \mathbf{v}_1,\mathbf{v}_2,\cdots,\mathbf{v}_{m} 与向量组 \mathbf{u}_1,\mathbf{u}_2,\cdots,\mathbf{u}_{m} 等价,再将正交向量组中每个向量经过单位化,就得到一个标准正交向量组,这种方法称为施密特正交化。

2 算法

S=\left\{\mathbf{v}_1, \ldots, \mathbf{v}_{m}\right\} 是某向量空间(\mathbb{R}^n)的基(其中 m \leq n)。由向量 \mathbf{v}_1, \ldots, \mathbf{v}_{m} 张成的子空间记为 \text{span}\left\{\mathbf{v}_1, \ldots, \mathbf{v}_{m}\right\}

那么可通过下列做法找到该向量空间中的 n 个两两正交的向量 S^{\prime}=\left\{\mathbf{u}_1, \ldots, \mathbf{u}_m\right\}。该方法称为施密特正交化(Gram–Schmidt process)。


具体方法就是以某一个基向量作为初始,将另一个向量依次向其投影,投影操作公式为:
\begin{equation} \operatorname{proj}_{\mathbf{u}}(\mathbf{v})=\frac{\langle\mathbf{u}, \mathbf{v}\rangle}{\langle\mathbf{u}, \mathbf{u}\rangle} \mathbf{u} \end{equation}

其中 \mathbf{u} \neq 0,如果 \mathbf{u} = 0 则记为:
\begin{equation} \operatorname {proj} _{\mathbf {0} }(\mathbf {v} ):=\mathbf {0} \end{equation}

根据上述方式,施密特正交化(Gram–Schmidt process)计算过程如下:

\begin{align} \mathbf{u}_1 & = \mathbf{v}_1, & \mathbf{e}_1 & = \frac{\mathbf{u}_1}{\|\mathbf{u}_1\|} \\ \mathbf{u}_2 & = \mathbf{v}_2-\operatorname{proj}_{\mathbf{u}_1} (\mathbf{v}_2), & \mathbf{e}_2 & = \frac{\mathbf{u}_2}{\|\mathbf{u}_2\|} \\ \mathbf{u}_3 & = \mathbf{v}_3-\operatorname{proj}_{\mathbf{u}_1} (\mathbf{v}_3) - \operatorname{proj}_{\mathbf{u}_2} (\mathbf{v}_3), & \mathbf{e}_3 & = \frac{\mathbf{u}_3 }{\|\mathbf{u}_3\|} \\ \mathbf{u}_4 & = \mathbf{v}_4-\operatorname{proj}_{\mathbf{u}_1} (\mathbf{v}_4)-\operatorname{proj}_{\mathbf{u}_2} (\mathbf{v}_4)-\operatorname{proj}_{\mathbf{u}_3} (\mathbf{v}_4), & \mathbf{e}_4 & = {\mathbf{u}_4 \over \|\mathbf{u}_4\|} \\ & {}\ \ \vdots & & {}\ \ \vdots \\ \mathbf{u}_{m} & = \mathbf{v}_{m} - \sum_{j=1}^{m-1}\operatorname{proj}_{\mathbf{u}_j} (\mathbf{v}_{m}), & \mathbf{e}_{m} & = \frac{\mathbf{u}_{m}}{\|\mathbf{u}_{m}\|}. \end{align}

这样就得到了 \text{span}\left\{\mathbf{v}_1, \ldots, \mathbf{v}_{m}\right\} 上的一组正交基 \left\{\mathbf{u}_1, \ldots, \mathbf{u}_m\right\},以及相应的标准正交基 \left\{\mathbf{e}_1, \ldots, \mathbf{e}_m\right\}

参考材料

[1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
[2] https://www.matongxue.com/parts/4664/

上一个 舒尔补 (Schur’s complement)
下一个 伴随 (Adjoints)

Add a Comment

Your email address will not be published.

Table of Contents