# 拉格朗日恒等式

{\displaystyle {\begin{aligned}{\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{aligned}}}

${\displaystyle \|\mathbf {a} \|^{2}\ \|\mathbf {b} \|^{2}-(\mathbf {a\cdot b} )^{2}=\sum _{1\leq i

${\displaystyle {\biggl (}\sum _{k=1}^{n}|a_{k}|^{2}{\biggr )}{\biggl (}\sum _{k=1}^{n}|b_{k}|^{2}{\biggr )}-{\biggl |}\sum _{k=1}^{n}a_{k}b_{k}{\biggr |}^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}|a_{i}{\overline {b}}_{j}-a_{j}{\overline {b}}_{i}|^{2}}$

## 拉格朗日恒等式和外代数

${\displaystyle (a\cdot a,b\cdot b)-(a\cdot b)^{2}=(a\wedge b)\cdot (a\wedge b)}$

${\displaystyle \|a\wedge b\|={\sqrt {(\|a\|\ \|b\|)^{2}-\|a\cdot b\|^{2}}}}$

## 参考资料

1. ^ Eric W. Weisstein. CRC concise encyclopedia of mathematics 2nd. CRC Press. 2003. ISBN 1-58488-347-2.
2. ^ Robert E Greene and Steven G Krantz. Exercise 16. Function theory of one complex variable 3rd. American Mathematical Society. 2006: 22. ISBN 0-8218-3962-4.
3. ^ Vladimir A. Boichenko, Gennadiĭ Alekseevich Leonov, Volker Reitmann. Dimension theory for ordinary differential equations. Vieweg+Teubner Verlag. 2005: 26. ISBN 3-519-00437-2.
4. ^ J. Michael Steele. Exercise 4.4: Lagrange's identity for complex numbers. The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities. Cambridge University Press. 2004: 68–69. ISBN 0-521-54677-X.
5. ^ Greene, Robert E.; Krantz, Steven G. Function Theory of One Complex Variable. Providence, R.I.: American Mathematical Society. 2002: 22, Exercise 16. ISBN 978-0-8218-2905-9.;
Palka, Bruce P. An Introduction to Complex Function Theory. Berlin, New York: Springer-Verlag. 1991: 27, Exercise 4.22. ISBN 978-0-387-97427-9.