微分¶
有矩阵 \(\mathbf{A} = [\mathbf{a}_1, \mathbf{a}_2, \cdots, \mathbf{a}_m]^{\top} \in \mathbb{R}^{m \times n}\)、向量 \(\mathbf{x} = [x_1, x_2, \cdots, x_n]^{\top} \in \mathbb{R}^{n}\) 和 \(\mathbf{y} = [y_1, y_2, \cdots, y_m]^{\top} \in \mathbb{R}^{m}\)，令：
\[\begin{split}
\begin{aligned}
\mathbf{f}:\; &\mathbb{R}^n \to \mathbb{R}^m\\
&\mathbf{x} \longmapsto \mathbf{y}
\end{aligned}
\end{split}\]
记 \(\mathbf{f} = (f_1, f_2, \cdots, f_m)^{\top}\)，其中：
\[\begin{split}
\begin{aligned}
f_i:\; &\mathbb{R}^n \to \mathbb{R}\\
&\mathbf{x} \longmapsto y_i
\end{aligned}
\end{split}\]
则有：
\[
\mathbf{df} = [\mathbf{d}f_1, \mathbf{d} f_2, \cdots, \mathbf{d} f_m]^{\top}
\]
\[\begin{split}
\mathbf{Ax} = \begin{bmatrix}
\langle \mathbf{a}_1, \mathbf{x} \rangle\\
\langle \mathbf{a}_2, \mathbf{x} \rangle\\
\vdots \\
\langle \mathbf{a}_m, \mathbf{x} \rangle
\end{bmatrix}
\end{split}\]
\(\nabla\) 定义¶
易知 \(\{\mathbf{d} x_1, \mathbf{d} x_2, \cdots, \mathbf{d} x_n\}\)，\(\{\mathbf{d} y_1, \mathbf{d} y_2, \cdots, \mathbf{d} y_m\}\) 分别为 \(\mathbb{R}^n\) 和 \(\mathbb{R}^m\) 的一组标准正交基，则记微分算子：
\[
\nabla_{\mathbf{x}} = (\frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, \cdots, \frac{\partial}{\partial x_n})^{\top},
\]
同样，也有：
\[
\nabla_{\mathbf{y}} = (\frac{\partial}{\partial y_1}, \frac{\partial}{\partial y_2}, \cdots, \frac{\partial}{\partial y_m})^{\top},
\]
则有
\[
\begin{cases}
\mathbf{dy}_i = \frac{\partial f_i(\mathbf{x})}{ \partial x_i} x_i
\end{cases}
\]
记微分导数：
\[
\frac{\partial \mathbf{f}} {\partial \mathbf{x}} = \mathbf{f}^{'}(\mathbf{x}) = \nabla \mathbf{f} = \frac{\partial (f_1, f_2, \cdots, f_m)} {\partial (x_1, x_2, \cdots, x_n)}
\]
这样，有：
\[
\mathbf{d} \mathbf{y} = \nabla_{\mathbf{x}}(\mathbf{y}) = \mathbf{f}^{'}(\mathbf{x}) \nabla_{\mathbf{x}} =
\]