矩阵中的求导

标量对向量求导

$$ y = f(x_1,\cdots,x_i,\cdots,x_n) $$

$$ X = [x_1,\cdots,x_i,\cdots,x_n] $$

$$ \frac {\partial y}{\partial X} = [\frac {\partial f}{\partial x_1},\cdots,\frac {\partial f}{\partial x_i},\cdots,\frac {\partial f}{\partial x_n}] $$

向量对向量求导

$$ Y = [f_1(x_1,\cdots,x_i,\cdots,x_n),\cdots, f_i(x_1,\cdots,x_i,\cdots,x_n),\cdots, f_m(x_1,\cdots,x_i,\cdots,x_n)] $$

$$ X = [x_1,\cdots,x_i,\cdots,x_n] $$

$$ \frac {\partial Y}{\partial X} = \begin{bmatrix} \frac {\partial f_1}{\partial x_1} & \frac {\partial f_2}{\partial x_1} & \cdots & \frac {\partial f_m}{\partial x_1}\cr \frac {\partial f_1}{\partial x_2} & \frac {\partial f_2}{\partial x_2} & \cdots & \frac {\partial f_m}{\partial x_1}\cr \vdots & \vdots & \ddots & \vdots\cr \frac {\partial f_1}{\partial x_n} & \cdots & \cdots & \frac {\partial f_m}{\partial x_n}\cr \end{bmatrix} $$

这是一个n行m列的矩阵,有时也会写成m行n列,都是一样的,区别在于加不加转置