本文摘译自 Wikipedia。
数学中，矩阵微积分是一种特殊的多元微积分表达形式，它把单个函数对多个变量或多元函数对单一变量表达为向量或矩阵的形式，使其可以视为一个整体作处理，这极大地简化了诸如多元函数极值和微分方程组等问题的求解过程。
表示法
在本文中，将采用如下所示的表示方法：
- $ \mathbf A, \mathbf X, \mathbf Y $ 等：粗体的大写字母，表示一个矩阵；
- $ \mathbf a, \mathbf x, \mathbf y $ 等：粗体的小写字母，表示一个向量；
- $ a, x, y $ 等：斜体的小写字母，表示一个标量；
- $ \mathbf X^T $：表示矩阵 $ \mathbf X $ 的转置；
- $ \mathbf X^H $：表示矩阵 $ \mathbf X $ 的共轭转置；
- $ | \mathbf X | $：表示方阵 $ \mathbf X $ 的行列式；
- $ || \mathbf x || $：表示向量 $ \mathbf x $ 的范数；
- $ \mathbf I $：表示单位矩阵。
向量微分
向量-标量
列向量函数 $ \mathbf y = \begin{bmatrix} y_1 & y_2 & \cdots & y_m \end{bmatrix}^T $ 对标量 $ x $ 的导数称为 $ \mathbf y $ 的切向量，可以以 分子记法 表示为
$ \frac{\partial \mathbf y}{\partial x} = \begin{bmatrix}
\frac{\partial y_1}{\partial x} \newline \frac{\partial y_2}{\partial x} \newline \vdots \newline \frac{\partial y_m}{\partial x}
\end{bmatrix}_{m \times 1} $
若以 分母记法 则可以表示为
$ \frac{\partial \mathbf y}{\partial x} = \begin{bmatrix}
\frac{\partial y_1}{\partial x} & \frac{\partial y_2}{\partial x} & \cdots & \frac{\partial y_m}{\partial x}
\end{bmatrix}_{1 \times m} $
标量-向量
标量函数 $ y $ 对列向量 $ \mathbf x = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}^T $ 的导数可以以 分子记法 表示为
$ \frac{\partial y}{\partial \mathbf x} = \begin{bmatrix}
\frac{\partial y}{\partial x_1} & \frac{\partial y}{\partial x_2} & \cdots & \frac{\partial y}{\partial x_n}
\end{bmatrix}_{1 \times n} $
若以 分母记法 则可以表示为
$ \frac{\partial y}{\partial \mathbf x} = \begin{bmatrix}
\frac{\partial y}{\partial x_1} \newline \frac{\partial y}{\partial x_2} \newline \vdots \newline \frac{\partial y}{\partial x_n}
\end{bmatrix}_{n \times 1} $
向量-向量
列向量函数 $ \mathbf y = \begin{bmatrix} y_1 & y_2 & \cdots & y_m \end{bmatrix}^T $ 对列向量 $ \mathbf x = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}^T $ 的导数可以以 分子记法 表示为
$ \frac{\partial \mathbf y}{\partial \mathbf x} = \begin{bmatrix}
\frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \cdots & \frac{\partial y_1}{\partial x_n} \newline
\frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_2}{\partial x_n} \newline
\vdots & \vdots & \ddots & \vdots \newline
\frac{\partial y_m}{\partial x_1} & \frac{\partial y_m}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_n} \newline
\end{bmatrix}_{m \times n} $
若以 分母记法 则可以表示为
$ \frac{\partial \mathbf y}{\partial \mathbf x} = \begin{bmatrix}
\frac{\partial y_1}{\partial x_1} & \frac{\partial y_2}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_1} \newline
\frac{\partial y_1}{\partial x_1} & \frac{\partial y_2}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_1} \newline
\vdots & \vdots & \ddots & \vdots \newline
\frac{\partial y_1}{\partial x_1} & \frac{\partial y_2}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_1} \newline
\end{bmatrix}_{n \times m} $
矩阵微分
矩阵-标量
形状为 $ m \times n $ 的矩阵函数 $ \mathbf Y $ 对标量 $ x $ 的导数称为 $ \mathbf Y $ 的切矩阵，可以以 分子记法 表示为
$ \frac{\partial \mathbf Y}{\partial x} = \begin{bmatrix}
\frac{\partial y_{11}}{\partial x} & \frac{\partial y_{12}}{\partial x} & \cdots & \frac{\partial y_{1n}}{\partial x} \newline
\frac{\partial y_{21}}{\partial x} & \frac{\partial y_{22}}{\partial x} & \cdots & \frac{\partial y_{2n}}{\partial x} \newline
\vdots & \vdots & \ddots & \vdots \newline
\frac{\partial y_{m1}}{\partial x} & \frac{\partial y_{m2}}{\partial x} & \cdots & \frac{\partial y_{mn}}{\partial x} \newline
\end{bmatrix}_{m \times n} $
标量-矩阵
标量函数 $ y $ 对形状为 $ p \times q $ 的矩阵 $ \mathbf X $ 的导数可以 分子记法 表示为
$ \frac{\partial y}{\partial \mathbf X} = \begin{bmatrix}
\frac{\partial y}{\partial x_{11}} & \frac{\partial y}{\partial x_{21}} & \cdots & \frac{\partial y}{\partial x_{p1}} \newline
\frac{\partial y}{\partial x_{12}} & \frac{\partial y}{\partial x_{22}} & \cdots & \frac{\partial y}{\partial x_{p2}} \newline
\vdots & \vdots & \ddots & \vdots \newline
\frac{\partial y}{\partial x_{1q}} & \frac{\partial y}{\partial x_{2q}} & \cdots & \frac{\partial y}{\partial x_{pq}} \newline
\end{bmatrix}_{q \times p} $
若以 分母记法 则可以表示为
$ \frac{\partial y}{\partial \mathbf X} = \begin{bmatrix}
\frac{\partial y}{\partial x_{11}} & \frac{\partial y}{\partial x_{12}} & \cdots & \frac{\partial y}{\partial x_{1q}} \newline
\frac{\partial y}{\partial x_{21}} & \frac{\partial y}{\partial x_{22}} & \cdots & \frac{\partial y}{\partial x_{2q}} \newline
\vdots & \vdots & \ddots & \vdots \newline
\frac{\partial y}{\partial x_{p1}} & \frac{\partial y}{\partial x_{p2}} & \cdots & \frac{\partial y}{\partial x_{pq}} \newline
\end{bmatrix}_{p \times q} $
恒等式
以下各式中，无特别备注，默认被求导的复合函数的各因式皆不是求导变量的函数。
向量-向量
| 表达式 | 分子记法 | 分母记法 | 备注 |
|---|---|---|---|
| $ \frac{\partial \mathbf a}{\partial \mathbf x} = $ | $ \mathbf 0 $ | $ \mathbf 0 $ | |
| $ \frac{\partial \mathbf x}{\partial \mathbf x} = $ | $ \mathbf I $ | $ \mathbf I $ | |
| $ \frac{\partial \mathbf A \mathbf x}{\partial \mathbf x} = $ | $ \mathbf A $ | $ \mathbf A^T $ | |
| $ \frac{\partial \mathbf x^T \mathbf A}{\partial \mathbf x} = $ | $ \mathbf A^T $ | $ \mathbf A $ | |
| $ \frac{\partial a \mathbf u}{\partial \mathbf x} = $ | $ a \frac{\partial \mathbf u}{\partial x} $ | $ a \frac{\partial \mathbf u}{\partial x} $ | $ \mathbf u = \mathbf u(\mathbf x) $ |
| $ \frac{\partial v \mathbf u}{\partial \mathbf x} = $ | $ v \frac{\partial \mathbf u}{\partial \mathbf x} + \mathbf u \frac{\partial v}{\partial \mathbf x} $ | $ v \frac{\partial \mathbf u}{\partial \mathbf x} + \frac{\partial v}{\partial \mathbf x} \mathbf u^T$ | $ v = v(\mathbf x), \mathbf u = \mathbf u(\mathbf x) $ |
| $ \frac{\partial \mathbf A \mathbf u}{\partial \mathbf x} = $ | $ \mathbf A \frac{\partial \mathbf u}{\partial \mathbf x} $ | $ \frac{\partial \mathbf u}{\partial \mathbf x} \mathbf A^T $ | $ \mathbf u = \mathbf u(\mathbf x) $ |
| $ \frac{\partial (\mathbf u + \mathbf v)}{\partial \mathbf x} = $ | $ \frac{\partial \mathbf u}{\partial \mathbf x} + \frac{\partial \mathbf v}{\partial \mathbf x} $ | $ \frac{\partial \mathbf u}{\partial \mathbf x} + \frac{\partial \mathbf v}{\partial \mathbf x} $ | $ \mathbf u = \mathbf u(\mathbf x), \mathbf v = \mathbf v(\mathbf x) $ |
| $ \frac{\partial \mathbf f(\mathbf g(\mathbf u))}{\partial \mathbf x} = $ | $ \frac{\partial \mathbf f(\mathbf g)}{\partial \mathbf g} \frac{\partial \mathbf g(\mathbf u)}{\partial \mathbf u} \frac{\partial \mathbf u}{\partial \mathbf x} $ | $ \frac{\partial \mathbf u}{\partial \mathbf x} \frac{\partial \mathbf g(\mathbf u)}{\partial \mathbf u} \frac{\partial \mathbf f(\mathbf g)}{\partial \mathbf g} $ | $ \mathbf u = \mathbf u(\mathbf x) $ |
标量-向量
| 表达式 | 分子记法 | 分母记法 | 备注 |
|---|---|---|---|
| $ \frac{\partial a}{\partial \mathbf x} = $ | $ \mathbf 0^T $ | $ \mathbf 0 $ | |
| $ \frac{\partial a u}{\partial \mathbf x} = $ | $ a \frac{\partial \mathbf u}{\partial \mathbf x} $ | $ a \frac{\partial \mathbf u}{\partial \mathbf x} $ | $ u = u(\mathbf x) $ |
| $ \frac{\partial (u + v)}{\partial \mathbf x} = $ | $ \frac{\partial u}{\partial \mathbf x} + \frac{\partial v}{\partial \mathbf x} $ | $ \frac{\partial u}{\partial \mathbf x} + \frac{\partial v}{\partial \mathbf x} $ | $ u = u(\mathbf x), v = v(\mathbf x) $ |
| $ \frac{\partial u v}{\partial \mathbf x} = $ | $ u \frac{\partial v}{\partial \mathbf x} + v \frac{\partial u}{\partial \mathbf x} $ | $ u \frac{\partial v}{\partial \mathbf x} + v \frac{\partial u}{\partial \mathbf x} $ | $ u = u(\mathbf x), v = v(\mathbf x) $ |
| $ \frac{\partial f(g(u))}{\partial \mathbf x} = $ | $ \frac{\partial f(g)}{\partial g} \frac{\partial g(u)}{\partial u} \frac{\partial u}{\partial \mathbf x} $ | $ \frac{\partial f(g)}{\partial g} \frac{\partial g(u)}{\partial u} \frac{\partial u}{\partial \mathbf x} $ | $ u = u(\mathbf x) $ |
| $ \frac{\partial (\mathbf u \cdot \mathbf v)}{\partial \mathbf x} = \frac{\partial \mathbf u^T \mathbf v}{\partial \mathbf x} = $ | $ \mathbf u^T \frac{\partial \mathbf v}{\partial \mathbf x} + \mathbf v^T \frac{\partial \mathbf u}{\partial \mathbf x} $ | $ \frac{\partial \mathbf v}{\partial \mathbf x} \mathbf u + \frac{\partial \mathbf u}{\partial \mathbf x} \mathbf v $ | $ \mathbf u = \mathbf u(\mathbf x), \mathbf v = \mathbf v(\mathbf x) $ |
| $ \frac{\partial (\mathbf u \cdot \mathbf A \mathbf v)}{\partial \mathbf x} = \frac{\partial \mathbf u^T \mathbf A \mathbf v}{\partial \mathbf x} = $ | $ \mathbf u^T \mathbf A \frac{\partial \mathbf v}{\partial \mathbf x} + \mathbf v^T \mathbf A^T \frac{\partial \mathbf u}{\partial \mathbf x} $ | $ \frac{\partial \mathbf v}{\partial \mathbf x} \mathbf A^T \mathbf u + \frac{\partial \mathbf u}{\partial \mathbf x} \mathbf A \mathbf v $ | $ \mathbf u = \mathbf u(\mathbf x), \mathbf v = \mathbf v(\mathbf x) $ |
| $ \frac{\partial (\mathbf a \cdot \mathbf u)}{\partial \mathbf x} = \frac{\partial \mathbf a^T \mathbf u}{\partial \mathbf x} = $ | $ \mathbf a^T \frac{\partial \mathbf u}{\partial \mathbf x} $ | $ \frac{\partial \mathbf u}{\partial \mathbf x} \mathbf a $ | $ \mathbf u = \mathbf u(\mathbf x) $ |
| $ \frac{\partial \mathbf b^T \mathbf A \mathbf x}{\partial \mathbf x} = $ | $ \mathbf b^T \mathbf A $ | $ \mathbf A^T \mathbf b $ | |
| $ \frac{\partial \mathbf x^T \mathbf A \mathbf x}{\partial \mathbf x} = $ | $ \mathbf x^T (\mathbf A + \mathbf A^T) $ | $ (\mathbf A + \mathbf A^T) \mathbf x $ | |
| $ \frac{\partial^2 \mathbf x^T \mathbf A \mathbf x}{\partial \mathbf x \partial \mathbf x^T} = $ | $ \mathbf A + \mathbf A^T $ | $ \mathbf A + \mathbf A^T $ | |
| $ \frac{\partial \mathbf a^T \mathbf x \mathbf x^T \mathbf b}{\partial \mathbf x} = $ | $ \mathbf x^T (\mathbf a \mathbf b^T + \mathbf b \mathbf a^T) $ | $ (\mathbf a \mathbf b^T + \mathbf b \mathbf a^T) \mathbf x $ | |
| $ \frac{\partial (\mathbf A \mathbf x + \mathbf b)^T \mathbf C (\mathbf D \mathbf x + \mathbf e)}{\partial \mathbf x} = $ | $ (\mathbf A \mathbf x + \mathbf b)^T \mathbf C \mathbf D + (\mathbf D \mathbf x + \mathbf e)^T \mathbf C^T \mathbf A $ | $ \mathbf D^T \mathbf C^T(\mathbf A \mathbf x + \mathbf b) + \mathbf A^T \mathbf C (\mathbf D \mathbf x + \mathbf e)^T $ | |
| $ \frac{\partial || \mathbf x ||^2}{\partial \mathbf x} = \frac{\partial (\mathbf x \cdot \mathbf x)}{\partial \mathbf x} = $ | $ 2 \mathbf x^T $ | $ 2 \mathbf x $ | |
| $ \frac{\partial || \mathbf x - \mathbf a || }{\partial \mathbf x} = $ | $ \frac{(\mathbf x - \mathbf a)^T}{ || \mathbf x - \mathbf a || } $ | $ \frac{(\mathbf x - \mathbf a)}{ || \mathbf x - \mathbf a || } $ |
向量-标量
| 表达式 | 分子记法 | 分母记法 | 备注 |
|---|---|---|---|
| $ \frac{\partial \mathbf a}{\partial x} = $ | $ \mathbf 0 $ | $ \mathbf 0 $ | |
| $ \frac{\partial a \mathbf u}{\partial x} = $ | $ a \frac{\partial \mathbf u}{\partial x} $ | $ a \frac{\partial \mathbf u}{\partial x} $ | $ \mathbf u = \mathbf u(\mathbf x) $ |
| $ \frac{\partial \mathbf A \mathbf u}{\partial x} = $ | $ \mathbf A \frac{\partial \mathbf u}{\partial x} $ | $ \frac{\partial \mathbf u}{\partial x} \mathbf A^T$ | $ \mathbf u = \mathbf u(\mathbf x) $ |
| $ \frac{\partial \mathbf u^T}{\partial x} = $ | $ \left( \frac{\partial \mathbf u}{\partial x} \right)^T $ | $ \left( \frac{\partial \mathbf u}{\partial x} \right)^T $ | $ \mathbf u = \mathbf u(\mathbf x) $ |
| $ \frac{\partial (\mathbf u + \mathbf v)}{\partial x} = $ | $ \frac{\partial \mathbf u}{\partial x} + \frac{\partial \mathbf v}{\partial x} $ | $ \frac{\partial \mathbf u}{\partial x} + \frac{\partial \mathbf v}{\partial x} $ | $ \mathbf u = \mathbf u(\mathbf x), \mathbf v = \mathbf v(\mathbf x) $ |
| $ \frac{\partial (\mathbf u^T \times \mathbf v)}{\partial x} = $ | $ \left( \frac{\partial \mathbf u}{\partial x} \right)^T \times \mathbf v + \mathbf u^T \times \frac{\partial \mathbf v}{\partial x} $ | $ \frac{\partial \mathbf u}{\partial x} \times \mathbf v + \mathbf u^T \times \left( \frac{\partial \mathbf v}{\partial x} \right)^T $ | $ \mathbf u = \mathbf u(\mathbf x), \mathbf v = \mathbf v(\mathbf x) $ |
| $ \frac{\partial \mathbf f(\mathbf g(\mathbf u))}{\partial x} = $ | $ \frac{\partial \mathbf f(\mathbf g)}{\partial \mathbf g} \frac{\partial \mathbf g(\mathbf u)}{\partial \mathbf u} \frac{\partial \mathbf u}{\partial x} $ | $\frac{\partial \mathbf u}{\partial x}\frac{\partial \mathbf g(\mathbf u)}{\partial \mathbf u} \frac{\partial \mathbf f(\mathbf g)}{\partial \mathbf g} $ | $ \mathbf u = \mathbf u(\mathbf x) $ |
| $ \frac{\partial (\mathbf U \times \mathbf v)}{\partial x} = $ | $ \frac{\partial \mathbf U}{\partial x} \times \mathbf v + \mathbf U \times \frac{\partial \mathbf v}{\partial x} $ | $ \mathbf v^T \times \frac{\partial \mathbf U}{\partial x} + \frac{\partial \mathbf v}{\partial x} \times \mathbf U^T $ | $ \mathbf U = \mathbf U(\mathbf x), \mathbf v = \mathbf v(\mathbf x) $ |
标量-矩阵
| 表达式 | 分子记法 | 分母记法 | 备注 |
|---|---|---|---|
| $ \frac{\partial a}{\partial \mathbf X} = $ | $ \mathbf 0^T $ | $ \mathbf 0 $ | |
| $ \frac{\partial a u}{\partial \mathbf X} = $ | $ a \frac{\partial u}{\partial \mathbf X} $ | $ a \frac{\partial u}{\partial \mathbf X} $ | $ u = u(\mathbf X) $ |
| $ \frac{\partial (u + v)}{\partial \mathbf X} = $ | $ \frac{\partial u}{\partial \mathbf X} + \frac{\partial v}{\partial \mathbf X} $ | $ \frac{\partial u}{\partial \mathbf X} + \frac{\partial v}{\partial \mathbf X} $ | $ u = u(\mathbf X), v = v(\mathbf X) $ |
| $ \frac{\partial u v}{\partial \mathbf X} = $ | $ u \frac{\partial v}{\partial \mathbf X} + v \frac{\partial u}{\partial \mathbf X} $ | $ u \frac{\partial v}{\partial \mathbf X} + v \frac{\partial u}{\partial \mathbf X} $ | $ u = u(\mathbf X), v = v(\mathbf X) $ |
| $ \frac{\partial f(g(u))}{\partial \mathbf X} = $ | $ \frac{\partial f(g)}{\partial g} \frac{\partial g(u)}{\partial u} \frac{\partial u}{\partial \mathbf X} $ | $ \frac{\partial f(g)}{\partial g} \frac{\partial g(u)}{\partial u} \frac{\partial u}{\partial \mathbf X} $ | $ u = u(\mathbf X) $ |
| $ \frac{\partial \mathbf a^T \mathbf X \mathbf b}{\partial \mathbf X} = $ | $ \mathbf b \mathbf a^T $ | $ \mathbf a \mathbf b^T $ | |
| $ \frac{\partial \mathbf a^T \mathbf X^T \mathbf b}{\partial \mathbf X} = $ | $ \mathbf a \mathbf b^T $ | $ \mathbf b \mathbf a^T $ | |
| $ \frac{\partial (\mathbf X \mathbf a + \mathbf b)^T \mathbf C (\mathbf X \mathbf a + \mathbf b)}{\partial \mathbf X} = $ | $ [ (\mathbf C + \mathbf C^T) (\mathbf X \mathbf a + \mathbf b) \mathbf a^T ]^T $ | $ (\mathbf C + \mathbf C^T) (\mathbf X \mathbf a + \mathbf b) \mathbf a^T $ | |
| $ \frac{\partial (\mathbf X \mathbf a)^T \mathbf C (\mathbf X \mathbf b)}{\partial \mathbf X} = $ | $ ( \mathbf C \mathbf X \mathbf b \mathbf a^T + \mathbf C^T \mathbf X \mathbf a \mathbf b^T )^T $ | $ \mathbf C \mathbf X \mathbf b \mathbf a^T + \mathbf C^T \mathbf X \mathbf a \mathbf b^T $ | |
| $ \frac{ \partial | \mathbf X | }{\partial \mathbf X} = $ | $ | \mathbf X | \mathbf X^{ - 1} $ | $ | \mathbf X | (\mathbf X^{ - 1})^T $ | |
| $ \frac{\partial \ln | a \mathbf X | }{\partial \mathbf X} = $ | $ \mathbf X^{ - 1} $ | $ (\mathbf X^{ - 1})^T $ | |
| $ \frac{ \partial | \mathbf A \mathbf X \mathbf B | }{\partial \mathbf X} = $ | $ | \mathbf A \mathbf X \mathbf B | \mathbf X^{ - 1} $ | $ | \mathbf A \mathbf X \mathbf B | (\mathbf X^{ - 1})^T $ | |
| $ \frac{ \partial | \mathbf X^n | }{\partial \mathbf X} = $ | $ n | \mathbf X^n | \mathbf X^{ - 1} $ | $ n | \mathbf X^n | (\mathbf X^{ - 1})^T $ | |
| $ \frac{ \partial \ln | \mathbf X^T \mathbf X | }{\partial \mathbf X} = $ | $ 2 \mathbf X^+ $ | $ 2 (\mathbf X^+)^T $ | $ \mathbf X^+ $ 为 $ \mathbf X $ 的广义逆 |
| $ \frac{\partial \ln | \mathbf X^T \mathbf X | }{\partial \mathbf X^+} = $ | $ - 2 \mathbf X $ | $ - 2 \mathbf X^T $ | $ \mathbf X^+ $ 为 $ \mathbf X $ 的广义逆 |
| $ \frac{\partial | \mathbf X^T \mathbf A \mathbf X | }{\partial \mathbf X} = $ | $ 2 | \mathbf X^T \mathbf A \mathbf X | \mathbf X^{ - 1} = 2 | \mathbf X^T | | \mathbf A | | \mathbf X | \mathbf X^{ - 1} $ | $ 2 | \mathbf X^T \mathbf A \mathbf X | (\mathbf X^{ - 1})^T $ | $ \mathbf X $ 为方阵且可逆 |
| $ \frac{\partial | \mathbf X^T \mathbf A \mathbf X | }{\partial \mathbf X} = $ | $ 2 | \mathbf X^T \mathbf A \mathbf X | ( \mathbf X^T \mathbf A^T \mathbf X )^{ - 1} \mathbf X^T \mathbf A^T $ | $ 2 | \mathbf X^T \mathbf A \mathbf X | \mathbf A \mathbf X ( \mathbf X^T \mathbf A \mathbf X )^{ - 1} $ | $ \mathbf A $ 对称 |
| $ \frac{\partial | \mathbf X^T \mathbf A \mathbf X | }{\partial \mathbf X} = $ | $ | \mathbf X^T \mathbf A \mathbf X | [ ( \mathbf X^T \mathbf A \mathbf X)^{ - 1} \mathbf X^T \mathbf A + ( \mathbf X^T \mathbf A^T \mathbf X )^{ - 1} \mathbf X^T \mathbf A^T ] $ | $ | \mathbf X^T \mathbf A \mathbf X | [ \mathbf A \mathbf X ( \mathbf X^T \mathbf A \mathbf X )^{ - 1} + \mathbf A^T \mathbf X ( \mathbf X^T \mathbf A^T \mathbf X )^{ - 1} ] $ |
矩阵-标量
| 表达式 | 分子记法 | 备注 |
|---|---|---|
| $ \frac{\partial a \mathbf U}{\partial x} = $ | $ a \frac{\partial \mathbf U}{\partial x} $ | $ \mathbf U = \mathbf U(x) $ |
| $ \frac{\partial \mathbf A \mathbf U \mathbf B}{\partial x} = $ | $ \mathbf A \frac{\partial \mathbf U}{\partial x} \mathbf B $ | $ \mathbf U = \mathbf U(x) $ |
| $ \frac{\partial (\mathbf U + \mathbf V)}{\partial x} = $ | $ \frac{\partial \mathbf U}{\partial x} + \frac{\partial \mathbf V}{\partial x} $ | $ \mathbf U = \mathbf U(x), \mathbf V = \mathbf V(x) $ |
| $ \frac{\partial (\mathbf U \mathbf V)}{\partial x} = $ | $ \mathbf U \frac{\partial \mathbf V}{\partial x} + \frac{\partial \mathbf U}{\partial x} \mathbf V $ | $ \mathbf U = \mathbf U(x), \mathbf V = \mathbf V(x) $ |
| $ \frac{\partial (\mathbf U \otimes \mathbf V)}{\partial x} = $ | $ \mathbf U \otimes \frac{\partial \mathbf V}{\partial x} + \frac{\partial \mathbf U}{\partial x} \otimes \mathbf V $ | $ \mathbf U = \mathbf U(x), \mathbf V = \mathbf V(x) $，$ \otimes $ 表示 Kronecker 乘积 |
| $ \frac{\partial (\mathbf U \circ \mathbf V)}{\partial x} = $ | $ \mathbf U \circ \frac{\partial \mathbf V}{\partial x} + \frac{\partial \mathbf U}{\partial x} \circ \mathbf V $ | $ \mathbf U = \mathbf U(x), \mathbf V = \mathbf V(x) $， $ \circ $ 表示 Hadamard 乘积 |
| $ \frac{\partial \mathbf U^{ - 1}}{\partial x} = $ | $ - \mathbf U^{ - 1} \frac{\partial \mathbf U}{\partial x} \mathbf U^{ - 1} $ | $ \mathbf U = \mathbf U(x) $ |
| $ \frac{\partial^2 \mathbf U^{ - 1}}{\partial x \partial y} = $ | $ \mathbf U^{ - 1} \left( \frac{\partial \mathbf U}{\partial x} \mathbf U^{ - 1} \frac{\partial \mathbf U}{\partial y} - \frac{\partial^2 \mathbf U}{\partial x \partial y} + \frac{\partial \mathbf U}{\partial y} \mathbf U^{ - 1} \frac{\partial \mathbf U}{\partial x} \right) \mathbf U^{ - 1} $ | $ \mathbf U = \mathbf U(x, y) $ |
| $ \frac{\partial g (x \mathbf A)}{\partial x} = $ | $ \mathbf A g’ (x \mathbf A) = g’ (x \mathbf A) \mathbf A $ | 应为 Hadamard 乘积；$ g (\cdot) $ 为逐元函数，如下例 |
| $ \frac{\partial e^{x \mathbf A}}{\partial x} = $ | $ \mathbf A e^{x \mathbf A} = e^{x \mathbf A} \mathbf A $ |
