点积

（重定向自Dot product

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

定义

代数定义

${\displaystyle {\vec {a}}\cdot {\vec {b}}=\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}}$

{\displaystyle {\begin{aligned}\ [1,3,-5]\cdot [4,-2,-1]&=(1)(4)+(3)(-2)+(-5)(-1)\\&=4-6+5\\&=3\end{aligned}}}

${\displaystyle {\vec {a}}\cdot {\vec {b}}={\vec {a}}{\vec {b}}^{T}}$

${\displaystyle {\begin{bmatrix}1&3&-5\end{bmatrix}}{\begin{bmatrix}4\\-2\\-1\end{bmatrix}}={\begin{bmatrix}3\end{bmatrix}}=3}$

几何定义

${\displaystyle {\vec {a}}\cdot {\vec {b}}=|{\vec {a}}|\,|{\vec {b}}|\cos \theta \;}$

${\displaystyle \cos {\theta }={\frac {\mathbf {a\cdot b} }{|{\vec {a}}|\,|{\vec {b}}|}}}$

标量投影

${\displaystyle A_{B}=|\mathbf {A} |\cos \theta }$

${\displaystyle \mathbf {A} \cdot \mathbf {B} =A_{B}|\mathbf {B} |=B_{A}|\mathbf {A} |}$

两种定义的等价性

由几何定义推出代数定义

${\displaystyle e_{1},...,e_{n}}$${\displaystyle \mathbb {R} ^{n}}$空间的一组标准正交基，可以得出：

{\displaystyle {\begin{aligned}\mathbf {A} &=[a_{1},\dots ,a_{n}]=\sum _{i}a_{i}\mathbf {e} _{i}\\\mathbf {B} &=[b_{1},\dots ,b_{n}]=\sum _{i}b_{i}\mathbf {e} _{i}.\end{aligned}}}

${\displaystyle \mathbf {A} \cdot \mathbf {B} =\mathbf {A} \cdot \sum _{i}b_{i}\mathbf {e} _{i}=\sum _{i}b_{i}(\mathbf {A} \cdot \mathbf {e} _{i})=\sum _{i}b_{i}a_{i}}$

由代数定义推出几何定义

${\displaystyle {\vec {v}}=v_{1}{\vec {i}}+v_{2}{\vec {j}}+v_{3}{\vec {k}}\;}$.

${\displaystyle |{\vec {v}}|^{2}=v_{1}^{2}+v_{2}^{2}+v_{3}^{2}\;}$.

${\displaystyle {\vec {v}}\cdot {\vec {v}}=v_{1}^{2}+v_{2}^{2}+v_{3}^{2}\;}$,

${\displaystyle {\vec {v}}\cdot {\vec {v}}=|{\vec {v}}|^{2}\;}$

${\displaystyle {\vec {c}}\equiv {\vec {a}}-{\vec {b}}\;}$,

${\displaystyle |{\vec {c}}|^{2}=|{\vec {a}}|^{2}+|{\vec {b}}|^{2}-2|{\vec {a}}||{\vec {b}}|\cos \theta \;}$.

${\displaystyle {\vec {c}}\cdot {\vec {c}}={\vec {a}}\cdot {\vec {a}}+{\vec {b}}\cdot {\vec {b}}-2|{\vec {a}}||{\vec {b}}|\cos \theta \;}$. （1）

${\displaystyle {\vec {c}}\cdot {\vec {c}}=({\vec {a}}-{\vec {b}})\cdot ({\vec {a}}-{\vec {b}})\;}$,

${\displaystyle {\vec {c}}\cdot {\vec {c}}={\vec {a}}\cdot {\vec {a}}+{\vec {b}}\cdot {\vec {b}}-2({\vec {a}}\cdot {\vec {b}})\;}$. （2）

${\displaystyle {\vec {a}}\cdot {\vec {a}}+{\vec {b}}\cdot {\vec {b}}-2({\vec {a}}\cdot {\vec {b}})={\vec {a}}\cdot {\vec {a}}+{\vec {b}}\cdot {\vec {b}}-2|{\vec {a}}||{\vec {b}}|\cos \theta \;}$.

${\displaystyle {\vec {a}}\cdot {\vec {b}}=|{\vec {a}}||{\vec {b}}|\cos \theta \;}$,

${\displaystyle \left\langle {\vec {a}},{\vec {b}}\right\rangle =\sum _{i=1}^{n}a_{i}b_{i}}$

性质

• 满足交换律
${\displaystyle {\vec {a}}\cdot {\vec {b}}={\vec {b}}\cdot {\vec {a}}}$
从定义即可证明（${\displaystyle \theta }$${\displaystyle a}$${\displaystyle b}$的夹角）：
${\displaystyle {\vec {a}}\cdot {\vec {b}}=\left\|{\vec {a}}\right\|\left\|{\vec {b}}\right\|\cos \theta =\left\|{\vec {b}}\right\|\left\|{\vec {a}}\right\|\cos \theta ={\vec {b}}\cdot {\vec {a}}}$
• 对向量加法满足分配律
${\displaystyle {\vec {a}}\cdot ({\vec {b}}+{\vec {c}})={\vec {a}}\cdot {\vec {b}}+{\vec {a}}\cdot {\vec {c}}}$
• 点积是双线性算子
${\displaystyle {\vec {a}}\cdot (r{\vec {b}}+{\vec {c}})=r({\vec {a}}\cdot {\vec {b}})+({\vec {a}}\cdot {\vec {c}})}$
• 乘以标量时满足：
${\displaystyle (c_{1}{\vec {a}})\cdot (c_{2}{\vec {b}})=(c_{1}c_{2})({\vec {a}}\cdot {\vec {b}})}$
• 不满足结合律。因为标量（${\displaystyle {\vec {a}}\cdot {\vec {b}}}$）与向量（${\displaystyle {\vec {c}}}$）的点积没有定义，所以结合律相关的表达式 ${\displaystyle ({\vec {a}}\cdot {\vec {b}})\cdot {\vec {c}}}$${\displaystyle {\vec {a}}\cdot ({\vec {b}}\cdot {\vec {c}})}$ 都没有良好的定义
• 两个非零向量${\displaystyle {\vec {a}}}$${\displaystyle {\vec {b}}}$正交的，当且仅当${\displaystyle {\vec {a}}\cdot {\vec {b}}=0}$

延伸

矩阵

${\displaystyle \mathbf {A} :\mathbf {B} =\sum _{i}\sum _{j}A_{ij}{\overline {B_{ij}}}=\mathrm {tr} (\mathbf {B} ^{\mathrm {H} }\mathbf {A} )=\mathrm {tr} (\mathbf {A} \mathbf {B} ^{\mathrm {H} }).}$

${\displaystyle \mathbf {A} :\mathbf {B} =\sum _{i}\sum _{j}A_{ij}B_{ij}=\mathrm {tr} (\mathbf {B} ^{\mathrm {T} }\mathbf {A} )=\mathrm {tr} (\mathbf {A} \mathbf {B} ^{\mathrm {T} })=\mathrm {tr} (\mathbf {A} ^{\mathrm {T} }\mathbf {B} )=\mathrm {tr} (\mathbf {B} \mathbf {A} ^{\mathrm {T} }).}$

参考文献

1. ^ 同济大学数学系 ．工程数学:线性代数(第六版)．高等教育出版社．2014