查看“︁拉格朗日恒等式”︁的源代码

在代数中，以[[约瑟夫·拉格朗日]]命名的'''拉格朗日恒等式'''是：<ref name=Weisstein>
{{cite book |title=CRC concise encyclopedia of mathematics |author=Eric W. Weisstein |year= 2003|url=http://books.google.com/?id=8LmCzWQYh_UC&pg=PA228 |isbn=1-58488-347-2 |edition=2nd |publisher=CRC Press }}</ref><ref name= Greene>{{cite book |title=Function theory of one complex variable |author=Robert E Greene and Steven G Krantz |url=http://www.amazon.com/Function-Complex-Variable-Graduate-Mathematics/dp/082182905X/ref=sr_1_1?ie=UTF8&s=books&qid=1271907834&sr=1-1#reader_082182905X |page=22 |chapter=Exercise 16   |isbn=0-8218-3962-4 |year=2006 |edition=3rd |publisher=American Mathematical Society}}</ref>

:<math>
\begin{align}
\biggl( \sum_{k=1}^n a_k^2\biggr) \biggl(\sum_{k=1}^n b_k^2\biggr) - \biggl(\sum_{k=1}^n a_k b_k\biggr)^2 & = \sum_{i=1}^{n-1} \sum_{j=i+1}^n(a_i b_j - a_j b_i)^2 \\
& \biggl(= \frac{1}{2} \sum_{i=1}^n \sum_{j=1,j\neq i}^n(a_i b_j - a_j b_i)^2\biggr),
\end{align}
</math>

应用于任意两个[[实数]]或[[复数 (数学)|复数]]集合（或者更一般地，一个[[交换环]]的元素）{''a''<sub>1</sub>, ''a''<sub>2</sub>, . . ., ''a<sub>n</sub>''} and {''b''<sub>1</sub>, ''b''<sub>2</sub>, . . ., ''b<sub>n</sub>''}。这个恒等式是[[婆罗摩笈多－斐波那契恒等式]]的推广，同时也是[[Binet–Cauchy恒等式]]的特殊形式。 

用一个更为简洁的[[向量]]形式[[表示論|表示]]，Lagrange恒等式就是：<ref name=Boichenko>{{cite book |title=Dimension theory for ordinary differential equations |author=Vladimir A. Boichenko, Gennadiĭ Alekseevich Leonov, Volker Reitmann |url=http://books.google.com/?id=9bN1-b_dSYsC&pg=PA26 |page=26 |isbn=3-519-00437-2 |year=2005 |publisher=Vieweg+Teubner Verlag}}</ref>

:<math>\| \mathbf a \|^2 \ \| \mathbf b \|^2 - (\mathbf {a \cdot b } )^2 = \sum_{1 \le i < j \le n} \left(a_ib_j-a_jb_i \right)^2 \ , </math>

其中'''a'''和'''b'''是由实数构成的''n''维向量。向复数的引申需要将[[点积]]理解为[[内积]]或者Hermitian点积。准确的说，对于复数，Lagrange恒等式可以写成以下形式：<ref name=Steele>{{cite book |title=The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities |author=J. Michael Steele |pages=68–69 |chapter=Exercise 4.4: Lagrange's identity for complex numbers |url=http://books.google.com/?id=bvgBdZKEYAEC&pg=PA68  |year=2004 |isbn=0-521-54677-X |publisher=Cambridge University Press}}</ref>

:<math>\biggl( \sum_{k=1}^n |a_k|^2\biggr) \biggl(\sum_{k=1}^n |b_k|^2\biggr) - \biggl|\sum_{k=1}^n a_k b_k\biggr|^2 = \sum_{i=1}^{n-1} \sum_{j=i+1}^n |a_i \overline{b}_j - a_j \overline{b}_i|^2</math>

用到复数的[[模]]<ref>
{{Cite book | last1=Greene | first1=Robert E. | last2=Krantz | first2=Steven G. | title=Function Theory of One Complex Variable | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-2905-9 | year=2002 | page=22, Exercise 16 | postscript=<!--None-->}};<br>
{{Cite book | last1=Palka | first1=Bruce P. | title=An Introduction to Complex Function Theory | url=https://archive.org/details/introductiontoco00palk | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-97427-9 | year=1991 | page=[https://archive.org/details/introductiontoco00palk/page/n45 27], Exercise 4.22 | postscript=<!--None-->}}
</ref>

== 拉格朗日恒等式和外代数 ==
拉格朗日恒等式用[[楔积]]可以写成
:<math>(a \cdot a,b \cdot b) -(a \cdot b)^2 =(a \wedge b) \cdot(a \wedge b)</math>。

因此，它可以看作是一个以点积形式给出两个向量楔积的公式，也就是由它们定义的平行四边形，即

:<math>\|a \wedge b\| = \sqrt{(\|a\|\ \|b\|)^2 - \|a \cdot b\|^2}</math>。

==参考资料==
{{reflist}}

{{約瑟夫·拉格朗日}}
[[Category:数学定理]]