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矩阵求导

符号​

  • 标量（Scalar）：xx
  • 向量（Vector）：x\mathbf{x}
  • 矩阵（Matrix）：X\mathbf{X}

布局​

  • Numerator Layout：分子布局，分子不变，分母转置
  • Denominator Layout：分母布局，分母不变，分子转置

向量对标量求导​

∂y∂x=[∂y1∂x∂y2∂x⋮∂ym∂x]\frac{\partial \mathbf{y}}{\partial x}=\begin{bmatrix} \frac{\partial y_1}{\partial x} \\ \frac{\partial y_2}{\partial x} \\ \vdots \\ \frac{\partial y_m}{\partial x} \end{bmatrix}

标量对向量求导​

∂y∂x=[∂y∂x1∂y∂x2⋯∂y∂xn]\frac{\partial y}{\partial \mathbf{x}}=\begin{bmatrix} \frac{\partial y}{\partial x_1} & \frac{\partial y}{\partial x_2} & \cdots & \frac{\partial y}{\partial x_n} \end{bmatrix}

向量对向量求导​

The vector-by-vector derivative is the Jacobian matrix of y\mathbf{y} with respect to x\mathbf{x}.

∂y∂x=[∂y1∂x1∂y1∂x2⋯∂y1∂xn∂y2∂x1∂y2∂x2⋯∂y2∂xn⋮⋮⋱⋮∂ym∂x1∂ym∂x2⋯∂ym∂xn]\frac{\partial \mathbf{y}}{\partial \mathbf{x}}=\begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \cdots & \frac{\partial y_1}{\partial x_n} \\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial y_m}{\partial x_1} & \frac{\partial y_m}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_n} \end{bmatrix}

where

y=[y1y2⋮ym],x=[x1x2⋮xn]\mathbf{y}=\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_m \end{bmatrix} ,\quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}

The bold symbols for vectors and matrices are not necessary, but they are used here for clarity.

矩阵对标量求导​

∂Y∂x=[∂y11∂x∂y12∂x⋯∂y1n∂x∂y21∂x∂y22∂x⋯∂y2n∂x⋮⋮⋱⋮∂ym1∂x∂ym2∂x⋯∂ymn∂x]\frac{\partial \mathbf{Y}}{\partial x}=\begin{bmatrix} \frac{\partial y_{11}}{\partial x} & \frac{\partial y_{12}}{\partial x} & \cdots & \frac{\partial y_{1n}}{\partial x} \\ \frac{\partial y_{21}}{\partial x} & \frac{\partial y_{22}}{\partial x} & \cdots & \frac{\partial y_{2n}}{\partial x} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial y_{m1}}{\partial x} & \frac{\partial y_{m2}}{\partial x} & \cdots & \frac{\partial y_{mn}}{\partial x} \end{bmatrix}

标量对矩阵求导​

∂y∂X=[∂y∂x11∂y∂x12⋯∂y∂x1n∂y∂x21∂y∂x22⋯∂y∂x2n⋮⋮⋱⋮∂y∂xm1∂y∂xm2⋯∂y∂xmn]\frac{\partial y}{\partial \mathbf{X}}=\begin{bmatrix} \frac{\partial y}{\partial x_{11}} & \frac{\partial y}{\partial x_{12}} & \cdots & \frac{\partial y}{\partial x_{1n}} \\ \frac{\partial y}{\partial x_{21}} & \frac{\partial y}{\partial x_{22}} & \cdots & \frac{\partial y}{\partial x_{2n}} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial y}{\partial x_{m1}} & \frac{\partial y}{\partial x_{m2}} & \cdots & \frac{\partial y}{\partial x_{mn}} \end{bmatrix}