论文中的一些数学符号表示_论文中ab是点乘还是乘法

向量点乘

向量的点乘，也叫向量的内积、数量积。结果是一个向量在另一个向量方向上投影的长度，是一个标量。

对于向量$a$和$b$，$A=\left[a_1,a_2,\dots a_n\right]$，$B=\left[b_1,b_2,\dots b_n\right]$

$$\mathrm{A}\cdot \mathbf{B}=\sum a_i{b}_i$$
或者：
$$\mathbf{A}\cdot \mathbf{B}=|\mathbf{A}\Vert \mathbf{B}|\cos \theta$$

向量叉乘

向量的叉乘，又叫向量积、外积、叉积。结果是一个和已有两个向量都垂直的向量。以三维向量为例

A × B = ∣ i j k a 1 a 2 a 3 b 1 b 2 b 3 ∣ = ( a 2 b 3 − b 2 a 3 ) i − ( a 1 b 3 − b 1 a 3 ) j + ( a 1 b 2 − b 1 a 2 ) k A\times B =\left|

$$\begin{array}{lll} i & j & k \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{array}$$ \right|=\left(a_{2} b_{3}-b_{2} a_{3} \right) i-\left(a_{1} b_{3}-b_{1}a_{3} \right) j+\left(a_{1} b_{2}-b_{1}a_{2}\right) k A×B=∣∣∣∣∣∣​ia1​b1​​ja2​b2​​ka3​b3​​∣∣∣∣∣∣​=(a2​b3​−b2​a3​)i−(a1​b3​−b1​a3​)j+(a1​b2​−b1​a2​)k

其中: $i=(1,0,0)\mathrm{j}=(0,1,0)\mathrm{k}=(0,0,1)$

张量（矩阵）点乘

张量（矩阵）的点乘，又叫哈达马积（Hadamard product），矩阵对应位置的元素相乘

$m\times n$ 矩阵 $A=\left[a_{ij}\right]$ 与 $m\times n$ 矩阵 $B=\left[b_{ij}\right]$ 的Hadamard积记作 $A\ast B$ 。

其元素定义为两个矩阵对应元素的乘积 $(A\ast B{)}_{ij}=a_{ij}{b}_{ij}$ ，例如
( 1 3 2 1 0 0 1 2 2 ) ∗ ( 0 0 2 7 5 0 2 1 1 ) = ( 1 ⋅ 0 3 ⋅ 0 2 ⋅ 2 1 ⋅ 7 0 ⋅ 5 0 ⋅ 0 1 ⋅ 2 2 ⋅ 1 2 ⋅ 1 ) = ( 0 0 4 7 0 0 2 2 2 ) \left(

$$\begin{array}{lll} 1 & 3 & 2 \\ 1 & 0 & 0 \\ 1 & 2 & 2 \end{array}$$ \right) *\left( $$\begin{array}{lll} 0 & 0 & 2 \\ 7 & 5 & 0 \\ 2 & 1 & 1 \end{array}$$ \right)=\left( $$\begin{array}{lll} 1 \cdot 0 & 3 \cdot 0 & 2 \cdot 2 \\ 1 \cdot 7 & 0 \cdot 5 & 0 \cdot 0 \\ 1 \cdot 2 & 2 \cdot 1 & 2 \cdot 1 \end{array}$$ \right)=\left( $$\begin{array}{lll} 0 & 0 & 4 \\ 7 & 0 & 0 \\ 2 & 2 & 2 \end{array}$$ \right) ⎝⎛​111​302​202​⎠⎞​∗⎝⎛​072​051​201​⎠⎞​=⎝⎛​1⋅01⋅71⋅2​3⋅00⋅52⋅1​2⋅20⋅02⋅1​⎠⎞​=⎝⎛​072​002​402​⎠⎞​

张量（矩阵）克罗内克乘积

克罗内克积是两个任意大小的矩阵间的运算，也叫直积或张量积。计算过程如下例所示:
( 1 2 3 1 ) ⊗ ( 0 3 2 1 ) = ( 1 ⋅ 0 1 ⋅ 3 2 ⋅ 0 2 ⋅ 3 1 ⋅ 2 1 ⋅ 1 2 ⋅ 2 2 ⋅ 1 3 ⋅ 0 3 ⋅ 3 1 ⋅ 0 1 ⋅ 3 3 ⋅ 2 3 ⋅ 1 1 ⋅ 2 1 ⋅ 1 ) = ( 0 3 0 6 2 1 4 2 0 9 0 3 6 3 2 1 ) \left(

$$\begin{array}{ll} 1 & 2 \\ 3 & 1 \end{array}$$ \right) \otimes\left( $$\begin{array}{ll} 0 & 3 \\ 2 & 1 \end{array}$$ \right)=\left( $$\begin{array}{cccc} 1 \cdot 0 & 1 \cdot 3 & 2 \cdot 0 & 2 \cdot 3 \\ 1 \cdot 2 & 1 \cdot 1 & 2 \cdot 2 & 2 \cdot 1 \\ 3 \cdot 0 & 3 \cdot 3 & 1 \cdot 0 & 1 \cdot 3 \\ 3 \cdot 2 & 3 \cdot 1 & 1 \cdot 2 & 1 \cdot 1 \end{array}$$ \right)=\left( $$\begin{array}{llll} 0 & 3 & 0 & 6 \\ 2 & 1 & 4 & 2 \\ 0 & 9 & 0 & 3 \\ 6 & 3 & 2 & 1 \end{array}$$ \right) (13​21​)⊗(02​31​)=⎝⎜⎜⎛​1⋅01⋅23⋅03⋅2​1⋅31⋅13⋅33⋅1​2⋅02⋅21⋅01⋅2​2⋅32⋅11⋅31⋅1​⎠⎟⎟⎞​=⎝⎜⎜⎛​0206​3193​0402​6231​⎠⎟⎟⎞​

拼接

张量（矩阵）的拼接可以按照不同的维度拼接

按照第一维度拼接：

( 1 2 3 1 ) ⊕ ( 0 3 2 1 ) = ( 1 2 0 3 3 1 2 1 ) \left(

$$\begin{array}{ll} 1 & 2 \\ 3 & 1 \end{array}$$ \right)\oplus\left( $$\begin{array}{ll} 0 & 3 \\ 2 & 1 \end{array}$$ \right)=\left( $$\begin{array}{llll} 1 & 2 & 0 & 3 \\ 3 & 1 & 2 & 1 \\ \end{array}$$ \right) (13​21​)⊕(02​31​)=(13​21​02​31​)

按照第二维度拼接：

( 1 2 3 1 ) ⊕ ( 0 3 2 1 ) = ( 1 2 3 1 0 3 2 1 ) \left(

$$\begin{array}{ll} 1 & 2 \\ 3 & 1 \end{array}$$ \right)\oplus\left( $$\begin{array}{ll} 0 & 3 \\ 2 & 1 \end{array}$$ \right)=\left( $$\begin{array}{llll} 1 & 2 \\ 3 & 1 \\ 0 & 3 \\ 2 & 1 \\ \end{array}$$ \right) (13​21​)⊕(02​31​)=⎝⎜⎜⎛​1302​2131​⎠⎟⎟⎞​

此外，数乘表示一个标量乘以一个矩阵或者向量中的每个元素