矩阵和向量求导规则

$\color{red}{本质是按元素求导，再按实际需求排列成想要的矩阵形状}$

$\color{#DC143C}{注:}$
$\color{#4285f4}{\ 1)内容从属于向量微积分}$
$\color{#4285f4}{2)标量使用小写字母表示，矩阵、向量均使用大写字母表示}$
$\color{#4285f4}{3)均采用分子布局，也称混合布局，即对向量与矩阵求导时需转置}$
$\color{#4285f4}{4)部分资料约定不一致，有争议的部分以相应资料为准}$

1. 对元素求导

1.1 矩阵对元素求导

$\frac {\partial{Y}}{\partial{x}} = \left[ \begin{matrix} \frac {\partial{y_{11}}}{\partial{x}} & \frac {\partial{y_{12}}}{\partial{x}} & \frac {\partial{y_{13}}}{\partial{x}} \\ \frac {\partial{y_{21}}}{\partial{x}} & \frac {\partial{y_{22}}}{\partial{x}} & \frac {\partial{y_{23}}}{\partial{x}} \\ \frac {\partial{y_{31}}}{\partial{x}} & \frac {\partial{y_{32}}}{\partial{x}} & \frac {\partial{y_{33}}}{\partial{x}} \end{matrix} \right]$

1.2 向量对元素求导

$\frac {\partial{Y^T}}{\partial{x}} = \left[ \begin{matrix} \frac {\partial{y_{1}}}{\partial{x}} & \frac {\partial{y_{2}}}{\partial{x}} & \frac {\partial{y_{3}}}{\partial{x}} \end{matrix} \right]$
$\frac {\partial{Y}}{\partial{x}} = \left[ \begin{matrix} \frac {\partial{y_{1}}}{\partial{x}} \\ \frac {\partial{y_{2}}}{\partial{x}} \\ \frac {\partial{y_{3}}}{\partial{x}} \end{matrix} \right]$

2. 对向量求导

2.1 元素对向量求导

$\frac {\partial{y}}{\partial{X}} = \left[ \begin{matrix} \frac {\partial{y}}{\partial{x_1}} & \frac {\partial{y}}{\partial{x_2}} & \frac {\partial{y}}{\partial{x_3}} \end{matrix} \right]$
$\frac {\partial{y}}{\partial{X^T}} = \left[ \begin{matrix} \frac {\partial{y}}{\partial{x_1}} \\ \frac {\partial{y}}{\partial{x_2}} \\ \frac {\partial{y}}{\partial{x_3}} \end{matrix} \right]$

2.2 向量对向量求导

$\frac {\partial{Y^T}}{\partial{X}} = \left[ \begin{matrix} \frac {\partial{Y^T}}{\partial{x_1}} & \frac {\partial{Y^T}}{\partial{x_2}} & \frac {\partial{Y^T}}{\partial{x_3}} \end{matrix} \right] , \frac {\partial{Y}}{\partial{X}} = \left[ \begin{matrix} \frac {\partial{Y}}{\partial{x_1}} & \frac {\partial{Y}}{\partial{x_2}} & \frac {\partial{Y}}{\partial{x_3}} \end{matrix} \right],$
$\frac {\partial{Y^T}}{\partial{X^T}} = \left[ \begin{matrix} \frac {\partial{Y^T}}{\partial{x_1}} \\ \frac {\partial{Y^T}}{\partial{x_2}} \\ \frac {\partial{Y^T}}{\partial{x_3}} \end{matrix} \right], \frac {\partial{Y}}{\partial{X^T}} = \left[ \begin{matrix} \frac {\partial{Y}}{\partial{x_1}} \\ \frac {\partial{Y}}{\partial{x_2}} \\ \frac {\partial{Y}}{\partial{x_3}} \end{matrix} \right]$

2.3 矩阵对向量求导

$\frac {\partial{Y}}{\partial{X}} = \left[ \begin{matrix} \frac {\partial{Y}}{\partial{x_1}} & \frac {\partial{Y}}{\partial{x_2}} & \frac {\partial{Y}}{\partial{x_3}} \end{matrix} \right]$
$\frac {\partial{Y}}{\partial{X^T}} = \left[ \begin{matrix} \frac {\partial{Y}}{\partial{x_1}} \\ \frac {\partial{Y}}{\partial{x_2}} \\ \frac {\partial{Y}}{\partial{x_3}} \end{matrix} \right]$

3. 对矩阵求导

3.1 元素对矩阵求导

$\frac {\partial{y}}{\partial{X}} = \left[ \begin{matrix} \frac {\partial{y}}{\partial{x_{11}}} & \frac {\partial{y}}{\partial{x_{21}}} & \frac {\partial{y}}{\partial{x_{31}}} \\ \frac {\partial{y}}{\partial{x_{12}}} & \frac {\partial{y}}{\partial{x_{22}}} & \frac {\partial{y}}{\partial{x_{32}}} \\ \frac {\partial{y}}{\partial{x_{13}}} & \frac {\partial{y}}{\partial{x_{23}}} & \frac {\partial{y}}{\partial{x_{33}}} \end{matrix} \right]$

3.2 向量对矩阵求导

$\frac {\partial{Y}}{\partial{X}} = \left[ \begin{matrix} \frac {\partial{Y}}{\partial{x_{11}}} & \frac {\partial{Y}}{\partial{x_{21}}} & \frac {\partial{Y}}{\partial{x_{31}}} \\ \frac {\partial{Y}}{\partial{x_{12}}} & \frac {\partial{Y}}{\partial{x_{22}}} & \frac {\partial{Y}}{\partial{x_{32}}} \\ \frac {\partial{Y}}{\partial{x_{13}}} & \frac {\partial{Y}}{\partial{x_{23}}} & \frac {\partial{Y}}{\partial{x_{33}}} \end{matrix} \right] , \frac {\partial{Y^T}}{\partial{X}} = \left[ \begin{matrix} \frac {\partial{Y^T}}{\partial{x_{11}}} & \frac {\partial{Y^T}}{\partial{x_{21}}} & \frac {\partial{Y^T}}{\partial{x_{31}}} \\ \frac {\partial{Y^T}}{\partial{x_{12}}} & \frac {\partial{Y^T}}{\partial{x_{22}}} & \frac {\partial{Y^T}}{\partial{x_{32}}} \\ \frac {\partial{Y^T}}{\partial{x_{13}}} & \frac {\partial{Y^T}}{\partial{x_{23}}} & \frac {\partial{Y^T}}{\partial{x_{33}}} \end{matrix} \right]$

3.3 矩阵对矩阵求导

$\color{#DC143C}{注：}\color{#4285f4}{有争议，以具体资料为准}$