# 几何代数

## 定义与符号

• ${\displaystyle AB\in {\mathcal {G}}(p,q)}$封闭
• ${\displaystyle 1A=A1=A}$，其中${\displaystyle 1}$是单位元（单位元的存在）
• ${\displaystyle A(BC)=(AB)C}$结合律
• ${\displaystyle A(B+C)=AB+AC}$ and ${\displaystyle (B+C)A=BA+CA}$分配律
• ${\displaystyle a^{2}=g(a,a)1}$，其中a是代数子空间V的任意元素。

### 几何积

${\displaystyle ab={\frac {1}{2}}(ab+ba)+{\frac {1}{2}}(ab-ba)}$

${\displaystyle a\cdot b:=g(a,b),}$

${\displaystyle {\frac {1}{2}}(ab+ba)={\frac {1}{2}}\left((a+b)^{2}-a^{2}-b^{2}\right)=a\cdot b}$

${\displaystyle a\wedge b:={\frac {1}{2}}(ab-ba)=-(b\wedge a)}$

${\displaystyle ab=a\cdot b+a\wedge b}$几何积的非广义或向量形式。

{\displaystyle {\begin{aligned}1\wedge a_{i}&=a_{i}\wedge 1=a_{i}\\a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r}&={\frac {1}{r!}}\sum _{\sigma \in {\mathfrak {S}}_{r}}\operatorname {sgn} (\sigma )a_{\sigma (1)}a_{\sigma (2)}\cdots a_{\sigma (r)},\end{aligned}}}

${\displaystyle a_{1}a_{2}a_{3}\dots a_{n}=\sum _{i=0}^{[{\frac {n}{2}}]}\sum _{\mu \in {}{\mathcal {C}}}(-1)^{k}Pf(a_{\mu _{1}}\cdot a_{\mu _{2}},\dots ,a_{\mu _{2i-1}}\cdot a_{\mu _{2i}})a_{\mu _{2i+1}}\land \dots \land a_{\mu _{n}}}$

### 刃、次、规范基

${\displaystyle [\mathbf {A} ]_{ij}=a_{i}\cdot a_{j}}$

${\displaystyle \sum _{k,l}[\mathbf {O} ]_{ik}[\mathbf {A} ]_{kl}[\mathbf {O} ^{\mathrm {T} }]_{lj}=\sum _{k,l}[\mathbf {O} ]_{ik}[\mathbf {O} ]_{jl}[\mathbf {A} ]_{kl}=[\mathbf {D} ]_{ij}}$

${\displaystyle e_{i}=\sum _{j}[\mathbf {O} ]_{ij}a_{j}}$

${\displaystyle {\begin{array}{rl}e_{1}e_{2}\cdots e_{r}&=e_{1}\wedge e_{2}\wedge \cdots \wedge e_{r}\\&=\left(\sum _{j}[\mathbf {O} ]_{1j}a_{j}\right)\wedge \left(\sum _{j}[\mathbf {O} ]_{2j}a_{j}\right)\wedge \cdots \wedge \left(\sum _{j}[\mathbf {O} ]_{rj}a_{j}\right)\\&=(\det \mathbf {O} )a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r}\end{array}}}$

${\displaystyle {\hat {e}}_{i}={\frac {1}{\sqrt {|e_{i}\cdot e_{i}|}}}e_{i},}$

${\displaystyle \{1,e_{1},e_{2},e_{3},e_{1}e_{2},e_{2}e_{3},e_{3}e_{1},e_{1}e_{2}e_{3}\}}$

${\displaystyle \left(\sum _{i}\alpha _{i}B_{i}\right)\left(\sum _{j}\beta _{j}B_{j}\right)=\sum _{i,j}\alpha _{i}\beta _{j}B_{i}B_{j}.}$

## 注释

1. ^ 几何代数的“外积”（outer product）与其他数学领域中的同名异义
2. ^ 给定${\textstyle u^{2}=1}$，可知${\textstyle ({\tfrac {1}{2}}(1+u))^{2}}$ ${\displaystyle ={\tfrac {1}{4}}(1+2u+uu)}$ ${\displaystyle ={\tfrac {1}{4}}(1+2u+1)}$ ${\displaystyle ={\tfrac {1}{2}}(1+u)}$，说明${\textstyle {\tfrac {1}{2}}(1+u)}$是幂等的，且${\displaystyle {\tfrac {1}{2}}(1+u)(1-u)}$ ${\displaystyle ={\tfrac {1}{2}}(1-uu)}$ ${\displaystyle ={\tfrac {1}{2}}(1-1)=0}$，表明它是非零零除子。
3. ^ 这是伪欧几里得向量空间标量积的同义词，指1-向量子空间上的对称双线性形式，而不是赋范向量空间上的内积。有人会将内积推广到整个代数，但实际上对此几乎没有共识。即使是有关几何代数的文章中，这个术语也不常用。
4. ^ 提到几何积下的分次时，文献一般只关注${\displaystyle \mathrm {Z} _{2}}$-分次，即分为奇数与偶数的${\displaystyle \mathrm {Z} }$-次。${\displaystyle \mathrm {Z} _{2}}$是几何积完整的${\displaystyle \mathrm {Z} _{2}{}^{n}}$-分次的一个子群。
5. ^ 次（grade）是齐性元素之次的同义词，是在作为代数的次与外积（${\displaystyle \mathrm {Z} }$-分次）下的次，而非在几何积下的次。[d]

## 外部链接

English translations of early books and papers

${\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}$