向量、标量、矩阵之间的求导

一.向量求导

1.向量求标量的导

Y = $\left[ \begin{matrix} y_1 \\ y_2 \\ \vdots\\ y_m \\ \end{matrix} \right]$ ，x是标量
那么：

\frac{\partial Y}{\partial x} = [\begin{matrix} \frac{\partial y_{1}}{x} \\ \frac{\partial y_{2}}{x} \\ ⋮ \\ \frac{\partial y_{m}}{x} \end{matrix}]

$\frac{\partial Y}{\partial x} = \left[ \begin{matrix} \frac{\partial y_1}{x} \\ \frac{\partial y_2}{x}\\ \vdots\\ \frac{\partial y_m}{x} \\ \end{matrix} \right]$

2.标量求向量的导

y是标量，X = $\left[ \begin{matrix} x_1 \\ x_2 \\ \vdots\\ x_n \\ \end{matrix} \right]$ ，
那么：

\frac{\partial y}{\partial X} = [\begin{matrix} \frac{\partial y}{\partial x_{1}} \\ \frac{\partial y}{\partial x_{2}} \\ ⋮ \\ \frac{\partial y}{\partial x_{n}} \end{matrix}]

$\frac{\partial y}{\partial X} = \left[ \begin{matrix} \frac{\partial y} {\partial x_1}\\ \frac{\partial y} {\partial x_2}\\ \vdots\\ \frac{\partial y} {\partial x_n} \\ \end{matrix} \right]$

3.向量求向量的导

Y = $\left[ \begin{matrix} y_1 \\ y_2 \\ \vdots\\ y_m \\ \end{matrix} \right]$ ， X = $\left[ \begin{matrix} x_1 \\ x_2 \\ \vdots\\ x_n \\ \end{matrix} \right]$ ，

那么：

\frac{\partial Y}{\partial X} = [\begin{matrix} \frac{\partial y_{1}}{\partial x_{1}} & \frac{\partial y_{1}}{\partial x_{2}} & \dots & \frac{\partial y_{1}}{\partial x_{n}} \\ \frac{\partial y_{2}}{\partial x_{1}} & \frac{\partial y_{2}}{\partial x_{2}} & \dots & \frac{\partial y_{2}}{\partial x_{n}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{\partial y_{m}}{\partial x_{1}} & \frac{\partial y_{m}}{\partial x_{2}} & \dots & \frac{\partial y_{m}}{\partial x_{n}} \end{matrix}]

$\frac{\partial Y}{\partial X} = \left[ \begin{matrix} \frac{\partial y_1} {\partial x_1} & \frac{\partial y_1} {\partial x_2} & \cdots & \frac{\partial y_1} {\partial x_n}\\ \frac{\partial y_2} {\partial x_1}& \frac{\partial y_2} {\partial x_2} & \cdots & \frac{\partial y_2} {\partial x_n}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial y_m} {\partial x_1} & \frac{\partial y_m} {\partial x_2} & \cdots & \frac{\partial y_m} {\partial x_n}\\ \end{matrix} \right]$

二. 矩阵求导规则

1.矩阵Y被标量x求导

Y是m行n列的矩阵， x是标量

\frac{\partial Y}{\partial x} = [\begin{matrix} \frac{\partial y_{11}}{\partial x} & \frac{\partial y_{12}}{\partial x} & \dots & \frac{\partial y_{1 n}}{\partial x} \\ \frac{\partial y_{21}}{\partial x} & \frac{\partial y_{22}}{\partial x} & \dots & \frac{\partial y_{2 n}}{\partial x} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{\partial y_{m 1}}{\partial x} & \frac{\partial y_{m 2}}{\partial x} & \dots & \frac{\partial y_{m n}}{\partial x} \end{matrix}]

$\frac{\partial Y}{\partial x} = \left[ \begin{matrix} \frac{\partial y_{11}} {\partial x} & \frac{\partial y_{12}} {\partial x} & \cdots & \frac{\partial y_{1n}} {\partial x}\\ \frac{\partial y_{21}} {\partial x}& \frac{\partial y_{22}} {\partial x} & \cdots & \frac{\partial y_{2n}} {\partial x}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial y_{m1}} {\partial x} & \frac{\partial y_{m2}} {\partial x} & \cdots & \frac{\partial y_{mn}} {\partial x}\\ \end{matrix} \right]$

3. 标量被矩阵求导

y是标量，X是p行q列的矩阵

\frac{\partial y}{\partial X} = [\begin{matrix} \frac{\partial y}{\partial x_{11}} & \frac{\partial y}{\partial x_{21}} & \dots & \frac{\partial y}{\partial x_{p 1}} \\ \frac{\partial y}{\partial x_{12}} & \frac{\partial y}{\partial x_{22}} & \dots & \frac{\partial y}{\partial x_{p 2}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{\partial y}{\partial x_{1 q}} & \frac{\partial y}{\partial x_{2 q}} & \dots & \frac{\partial y}{\partial x_{p q}} \end{matrix}]

$\frac{\partial y}{\partial X} = \left[ \begin{matrix} \frac{\partial y} {\partial x_{11}} & \frac{\partial y} {\partial x_{21}} & \cdots & \frac{\partial y} {\partial x_{p1}}\\ \frac{\partial y} {\partial x_{12}}& \frac{\partial y} {\partial x_{22}} & \cdots & \frac{\partial y} {\partial x_{p2}}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial y} {\partial x_{1q}} & \frac{\partial y} {\partial x_{2q}} & \cdots & \frac{\partial y} {\partial x_{pq}}\\ \end{matrix} \right]$
注意求导后的矩阵是q x p大小的矩阵