【数学】矩阵导数

基本说明

• 矩阵求导无非就是：一个矩阵里的元素对另一个矩阵里的元素进行对应求导，只要符合一些规则而已;这些规则就是：
  • 分子布局(Numerator layout)：即分子为列向量，分母为行向量
  • 分母布局(Denominator layout)：即分子为行向量,分母为列向量
• 分子布局(Numerator layout)
  • 假设：x,y为常数$$x=[x_1,x_2,\dots ,x_n{]}^T$$$$y=[y_1,y_2,\dots ,y_m{]}^T$$$$X=\left[\begin{array}{ccc}{x}_{11},x_{12}& \cdots & x_{1n}\\ x_{21},x_{22}& \cdots & x_{2n}\\ ⋮,⋮& \ddots & ⋮\\ x_{n1},x_{n2}& \cdots & x_{nn}\end{array}\right]$$$$Y=\left[\begin{array}{ccc}{y}_{11},y_{12}& \cdots & y_{1n}\\ y_{21},y_{22}& \cdots & y_{2n}\\ ⋮,⋮& \ddots & ⋮\\ y_{n1},y_{n2}& \cdots & y_{nn}\end{array}\right]$$
• 向量/标量 ：$$\frac{\partial y}{\partial x}=[\frac{\partial y_1}{\partial x},\frac{\partial y_2}{\partial x},...,\frac{\partial y_m}{\partial x}{]}^T$$
• 标量/向量：$$\frac{\partial y}{\partial x}=[\frac{\partial y}{\partial x_1},\frac{\partial y}{\partial x_2},...,\frac{\partial y}{\partial x_n}]$$
• 向量/向量：
  
  这就是著名的雅可比（Jacobian）矩阵
• 矩阵/标量：
  
• 标量/矩阵：
  
• 矩阵/向量：
  
• 向量/矩阵：$$\frac{\partial (x^T)}{\partial Y}=\left[\begin{array}{ccc}\frac{\partial x_1}{\partial y_{11}},\frac{\partial x_2}{\partial y_{21}}& \cdots & \frac{\partial x_n}{\partial y_{n1}}\\ & & \\ \frac{\partial x_1}{\partial y_{12}},\frac{\partial x_2}{\partial y_{22}}& \cdots & \frac{\partial x_n}{\partial y_{n2}}\\ & & \\ ⋮,⋮& \ddots & ⋮\\ & & \\ \frac{\partial x_1}{\partial y_{1n}},\frac{\partial x_2}{\partial y_{2n}}& \cdots & \frac{\partial x_n}{\partial y_{nn}}\end{array}\right]$$
• 矩阵/矩阵：$$\frac{\partial Y}{\partial X}=\left[\begin{array}{ccc}\frac{\partial y_{11}}{\partial x_{11}},\frac{\partial y_{12}}{\partial x_{21}}& \cdots & \frac{\partial y_{1n}}{\partial x_{n1}}\\ & & \\ \frac{\partial y_{21}}{\partial x_{12}},\frac{\partial y_{22}}{\partial x_{22}}& \cdots & \frac{\partial y_{2n}}{\partial y_{n2}}\\ & & \\ ⋮,⋮& \ddots & ⋮\\ & & \\ \frac{\partial y_{n1}}{\partial x_{1n}},\frac{\partial y_{n2}}{\partial x_{2n}}& \cdots & \frac{\partial y_{nn}}{\partial y_{nn}}\end{array}\right]$$
• 分母布局(Denominator layout)
  • 就是上面所有结果的转置。

运算公式

• 知道啥叫分子布局和分母布局了，然后对于要求导的式子，就可以对照着快速查阅下面的表格了:
  
  
  
  
  
  
  
  
  
  参考链接：https://yq.aliyun.com/articles/411340