矩阵求导

行向量\(y^T=\{1,2,3,...,n\}\)和 列向量 \[y=\begin{bmatrix} 1 \\ \vdots \\ n \end{bmatrix}\]

的表示，这里设计到一部分Latex的符号，见LaTeX 各种命令，符号 - CSDN博客 。

举一个例子

\[Y = \begin{bmatrix} y_{11} & \cdots & y_{1n} \\ \vdots & \ddots & \vdots \\ y_{m1} & \cdots & y_{mn} \end{bmatrix} = \begin{bmatrix} y_{1}^T \\ \vdots \\ y_{m}^T \\ \end{bmatrix}\]是一个\(m \times n\)矩阵， \[X = \begin{bmatrix} x_{11} & \cdots & x_{1q} \\ \vdots & \ddots & \vdots \\ x_{p1} & \cdots & x_{pq} \end{bmatrix} = \begin{bmatrix}x_1,x_2,\cdots,x_q\end{bmatrix}\]

因此，

\[ \frac{\partial Y}{\partial X}= \begin{bmatrix}\frac{\partial Y}{\partial x_1}, \cdots, \frac{\partial Y}{\partial x_q}\end{bmatrix} =\begin{bmatrix} \frac{\partial y_{1}^T}{\partial X} \\ \vdots \\ \frac{\partial y_{m}^T}{\partial X}\\ \end{bmatrix} =\begin{bmatrix} \frac{\partial y_{1}^T}{\partial x_1} & \cdots & \frac{\partial y_{1}^T}{\partial x_q} \\ \vdots & \ddots & \vdots \\ \frac{\partial y_{1}^T}{\partial x_1} & \cdots & \frac{\partial y_{1}^T}{\partial x_q} \\ \end{bmatrix} \]