格拉姆-施密特正交化

• ${\displaystyle {\boldsymbol {V}}^{n}}$：维数为n 的内积空间
• ${\displaystyle {\boldsymbol {v}}\in {\boldsymbol {V}}^{n}}$：${\displaystyle {\boldsymbol {V}}^{n}}$中的元素，可以是向量、函数，等等
• ${\displaystyle \langle {\boldsymbol {v}}_{1},{\boldsymbol {v}}_{2}\rangle }$：${\displaystyle {\boldsymbol {v}}_{1}}$与${\displaystyle {\boldsymbol {v}}_{2}}$的内积
• ${\displaystyle \mathrm {span} \{{\boldsymbol {v}}_{1},{\boldsymbol {v}}_{2},\ldots ,{\boldsymbol {v}}_{n}\}}$：${\displaystyle {\boldsymbol {v}}_{1}}$、${\displaystyle {\boldsymbol {v}}_{2}}$……${\displaystyle {\boldsymbol {v}}_{n}}$张成的子空间
• ${\displaystyle \mathrm {proj} _{\boldsymbol {v}}\,{\boldsymbol {u}}={\langle {\boldsymbol {u}},{\boldsymbol {v}}\rangle \over \langle {\boldsymbol {v}},{\boldsymbol {v}}\rangle }{\boldsymbol {v}}}$：${\displaystyle {\boldsymbol {u}}}$在${\displaystyle {\boldsymbol {v}}}$上的投影

图1 ${\displaystyle {\boldsymbol {v}}}$在${\displaystyle {\boldsymbol {V}}^{2}}$上投影，构造${\displaystyle {\boldsymbol {V}}^{3}}$上的正交基${\displaystyle {\boldsymbol {\beta }}}$

Gram-Schmidt正交化的基本想法，是利用投影原理在已有正交基的基础上构造一个新的正交基。

设${\displaystyle {\boldsymbol {v}}\in {\boldsymbol {V^{n}}}}$。${\displaystyle {\boldsymbol {V}}^{k}}$是${\displaystyle {\boldsymbol {V}}^{n}}$上的${\displaystyle k}$维子空间，其标准正交基为${\displaystyle \{{\boldsymbol {\eta }}_{1},\ldots ,{\boldsymbol {\eta }}_{k}\}}$，且${\displaystyle {\boldsymbol {v}}}$不在${\displaystyle {\boldsymbol {V}}^{k}}$上。由投影原理知，${\displaystyle {\boldsymbol {v}}}$与其在${\displaystyle {\boldsymbol {V}}^{k}}$上的投影${\displaystyle \mathrm {proj} _{\boldsymbol {V^{k}}}{\boldsymbol {v}}}$之差

${\displaystyle {\boldsymbol {\beta }}={\boldsymbol {v}}-\sum _{i=1}^{k}\mathrm {proj} _{{\boldsymbol {\eta }}_{i}}\,{\boldsymbol {v}}={\boldsymbol {v}}-\sum _{i=1}^{k}\langle {\boldsymbol {v}},{\boldsymbol {\eta }}_{i}\rangle {\boldsymbol {\eta }}_{i}}$

是正交于子空间${\displaystyle {\boldsymbol {V}}^{k}}$的，亦即${\displaystyle {\boldsymbol {\beta }}}$正交于${\displaystyle {\boldsymbol {V}}^{k}}$的正交基${\displaystyle {\boldsymbol {\eta }}_{i}}$。因此只要将${\displaystyle {\boldsymbol {\beta }}}$单位化，即

${\displaystyle {\boldsymbol {\eta }}_{k+1}={\frac {\boldsymbol {\beta }}{\|{\boldsymbol {\beta }}\|}}={\frac {\boldsymbol {\beta }}{\sqrt {\langle {\boldsymbol {\beta }},{\boldsymbol {\beta }}\rangle }}}}$

那么${\displaystyle \{{\boldsymbol {\eta }}_{1},\ldots ,{\boldsymbol {\eta }}_{k},{\boldsymbol {\eta }}_{k+1}\}}$就是${\displaystyle {\boldsymbol {V}}^{k}}$在${\displaystyle {\boldsymbol {v}}}$上扩展的子空间${\displaystyle \mathrm {span} \{{\boldsymbol {v}},{\boldsymbol {\eta }}_{1},...,{\boldsymbol {\eta }}_{k}\}}$的标准正交基。

根据上述分析，对于向量组${\displaystyle \{{\boldsymbol {v}}_{1},\ldots ,{\boldsymbol {v}}_{m}\}}$张成的空间${\displaystyle {\boldsymbol {V}}^{m}}$ (${\displaystyle m<n}$)，只要从其中一个向量（不妨设为${\displaystyle {\boldsymbol {v}}_{1}}$）所张成的一维子空间${\displaystyle \mathrm {span} \{{\boldsymbol {v}}_{1}\}}$开始（注意到${\displaystyle {\boldsymbol {v}}_{1}}$就是${\displaystyle \mathrm {span} \{{\boldsymbol {v}}_{1}\}}$的正交基），重复上述扩展构造正交基的过程，就能够得到${\displaystyle {\boldsymbol {V}}^{n}}$ 的一组正交基。这就是Gram-Schmidt正交化。

首先需要确定已有基底向量的顺序，不妨设为${\displaystyle \{{\boldsymbol {v}}_{1},\ldots ,{\boldsymbol {v}}_{n}\}}$。Gram-Schmidt正交化的过程如下：

	${\displaystyle {\boldsymbol {\beta }}_{1}={\boldsymbol {v}}_{1},}$		${\displaystyle {\boldsymbol {\eta }}_{1}={{\boldsymbol {\beta }}_{1} \over \|{\boldsymbol {\beta }}_{1}\|}}$
	${\displaystyle {\boldsymbol {\beta }}_{2}={\boldsymbol {v}}_{2}-\langle {\boldsymbol {v}}_{2},{\boldsymbol {\eta }}_{1}\rangle {\boldsymbol {\eta }}_{1},}$		${\displaystyle {\boldsymbol {\eta }}_{2}={{\boldsymbol {\beta }}_{2} \over \|{\boldsymbol {\beta }}_{2}\|}}$
	${\displaystyle {\boldsymbol {\beta }}_{3}={\boldsymbol {v}}_{3}-\langle {\boldsymbol {v}}_{3},{\boldsymbol {\eta }}_{1}\rangle {\boldsymbol {\eta }}_{1}-\langle {\boldsymbol {v}}_{3},{\boldsymbol {\eta }}_{2}\rangle {\boldsymbol {\eta }}_{2},}$		${\displaystyle {\boldsymbol {\eta }}_{3}={{\boldsymbol {\beta }}_{3} \over \|{\boldsymbol {\beta }}_{3}\|}}$
	${\displaystyle \vdots }$		${\displaystyle \vdots }$
	${\displaystyle {\boldsymbol {\beta }}_{n}={\boldsymbol {v}}_{n}-\sum _{i=1}^{n-1}\langle {\boldsymbol {v}}_{n},{\boldsymbol {\eta }}_{i}\rangle {\boldsymbol {\eta }}_{i},}$		${\displaystyle {\boldsymbol {\eta }}_{n}={{\boldsymbol {\beta }}_{n} \over \|{\boldsymbol {\beta }}_{n}\|}}$

这样就得到${\displaystyle \mathrm {span} \{{\boldsymbol {v}}_{1},\ldots ,{\boldsymbol {v}}_{n}\}}$上的一组正交基${\displaystyle \{{\boldsymbol {\beta }}_{1},\ldots ,{\boldsymbol {\beta }}_{n}\}}$，以及相应的标准正交基${\displaystyle \{{\boldsymbol {\eta }}_{1},\ldots ,{\boldsymbol {\eta }}_{n}\}}$。

例

考察如下欧几里得空间Rⁿ中向量的集合，欧氏空间上内积的定义为<a, b> = b^Ta：

${\displaystyle S=\lbrace {\boldsymbol {v}}_{1}={\begin{pmatrix}3\\1\end{pmatrix}},{\boldsymbol {v}}_{2}={\begin{pmatrix}2\\2\end{pmatrix}}\rbrace .}$

下面作Gram－Schmidt正交化，以得到一组正交向量：

${\displaystyle {\boldsymbol {\beta }}_{1}={\boldsymbol {v}}_{1}={\begin{pmatrix}3\\1\end{pmatrix}}}$

${\displaystyle {\boldsymbol {\beta }}_{2}={\boldsymbol {v}}_{2}-\mathrm {proj} _{{\boldsymbol {\beta }}_{1}}\,{\boldsymbol {v}}_{2}={\begin{pmatrix}2\\2\end{pmatrix}}-\mathrm {proj} _{\begin{pmatrix}3\\1\end{pmatrix}}\,{\begin{pmatrix}2\\2\end{pmatrix}}={\begin{pmatrix}-2/5\\6/5\end{pmatrix}}}$

下面验证向量${\displaystyle {\boldsymbol {\beta }}_{1}}$与${\displaystyle {\boldsymbol {\beta }}_{2}}$的正交性：

${\displaystyle \langle {\boldsymbol {\beta }}_{1},{\boldsymbol {\beta }}_{2}\rangle =\left\langle {\begin{pmatrix}3\\1\end{pmatrix}},{\begin{pmatrix}-2/5\\6/5\end{pmatrix}}\right\rangle =-{\frac {6}{5}}+{\frac {6}{5}}=0.}$

将这些向量单位化：

${\displaystyle {\boldsymbol {\eta }}_{1}={1 \over {\sqrt {10}}}{\begin{pmatrix}3\\1\end{pmatrix}}}$

${\displaystyle {\boldsymbol {\eta }}_{2}={1 \over {\sqrt {8 \over 5}}}{\begin{pmatrix}-2/5\\6/5\end{pmatrix}}}$

于是${\displaystyle \{{\boldsymbol {\eta }}_{1},{\boldsymbol {\eta }}_{2}\}}$就是 ${\displaystyle \mathrm {span} \{{\boldsymbol {v}}_{1},{\boldsymbol {v}}_{2}\}}$ 的一组标准正交基底。

随着内积空间上内积的定义以及构成内积空间的元素的不同，Gram-Schmidt正交化也表现出不同的形式。

例如，在实向量空间上，内积定义为：

${\displaystyle \langle {\boldsymbol {a}},{\boldsymbol {b}}\rangle ={\boldsymbol {b}}^{T}{\boldsymbol {a}}}$

在复向量空间上，内积定义为：

${\displaystyle \langle {\boldsymbol {a}},{\boldsymbol {b}}\rangle ={\boldsymbol {b}}^{H}{\boldsymbol {a}}}$

函数之间的内积则定义为：

${\displaystyle \langle f(x),g(x)\rangle =\int _{-\infty }^{\infty }f(x)g(x)dx}$

与之对应，相应的Gram－Schmidt正交化就具有不同的形式。

• 内积空间
• 内积
• 正交
• QR分解

• Hazewinkel, Michiel (编), , , Springer, 2001, ISBN 978-1-55608-010-4
• Harvey Mudd College Math Tutorial on the Gram-Schmidt algorithm
• Earliest known uses of some of the words of mathematics: G （页面存档备份，存于）
• Demos: Gram Schmidt process in plane and Gram Schmidt process in space
• Gram-Schmidt orthogonalization applet （页面存档备份，存于）
• NAG Gram–Schmidt orthogonalization of n vectors of order m routine （页面存档备份，存于）
• Proof: Raymond Puzio, Keenan Kidwell. "proof of Gram-Schmidt orthogonalization algorithm" (version 8). PlanetMath.org. （页面存档备份，存于）