标签：常用 right mathbf 运算 矩阵 alpha frac partial 21

以下内容是把这里的结论整理了一下。

矩阵乘法

1. 已知\(\mathbf{C}=\mathbf{A}\mathbf{B}\)，则：

   \[\mathbf{C}^T=\mathbf{B}^T\mathbf{A}^T \]

2. 已知\(\mathbf{C}=\mathbf{A}\mathbf{B}\)，则：

   \[\mathbf{C}^{-1}=\mathbf{B}^{-1}\mathbf{A}^{-1} \]

分块矩阵

已知\(\mathbf{A}=\begin{bmatrix}\mathbf{A_{11}}&\mathbf{A_{12}}\\\mathbf{A_{21}}&\mathbf{A_{22}}\end{bmatrix}\)，则：

\[\mathbf{A}^{-1}=\left[\begin{array}{cc} \left(\mathbf{A}_{11}-\mathbf{A}_{12} \mathbf{A}_{22}^{-1} \mathbf{A}_{21}\right)^{-1} & -\mathbf{A}_{11}^{-1} \mathbf{A}_{12}\left(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\right)^{-1} \\ -\mathbf{A}_{22}^{-1} \mathbf{A}_{21}\left(\mathbf{A}_{11}-\mathbf{A}_{12} \mathbf{A}_{22}^{-1} \mathbf{A}_{21}\right)^{-1} & \left(\mathbf{A}_{22}-\mathbf{A}_{21} \mathbf{A}_{11}^{-1} \mathbf{A}_{12}\right)^{-1} \end{array}\right] \]

向量微分

向量对向量求微分（雅克比矩阵）的定义：

\(y\)是一个有\(m\)个元素的向量，\(x\)是一个有\(n\)个元素的向量，则雅克比矩阵定义为：

\[\frac{\partial \mathbf{y}}{\partial \mathbf{x}}=\left[\begin{array}{cccc} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \cdots & \frac{\partial y_1}{\partial x_n} \\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_2}{\partial x_n} \\ \vdots & \vdots & & \vdots \\ \frac{\partial y_m}{\partial x_1} & \frac{\partial y_m}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_n} \end{array}\right] \]

下列表达式中均有\(\mathbf{A}\)与\(\mathbf{x}\)无关。

1. 已知\(\mathbf{y}=\mathbf{A}\mathbf{x}\)，则

   \[\frac{\partial \mathbf{y}}{\partial \mathbf{x}}=\mathbf{A} \]
   • 令\(\mathbf A=\mathbf I\)，还可以知道\(\frac{\partial \mathbf{x}}{\partial \mathbf{x}}=\mathbf{I}\)

2. 已知\(\mathbf{x}\)是\(\mathbf{z}\)的函数，\(\mathbf{y}=\mathbf{A}\mathbf{x}\)，则：

   \[\frac{\partial \mathbf{y}}{\partial \mathbf{z}}=\mathbf{A}\frac{\partial \mathbf{x}}{\partial \mathbf{z}} \]

3. 已知\(\mathbf y=\mathbf{x}^{T}\mathbf{A}\mathbf{x}\)，则：

   \[\frac{\partial \mathbf y}{\partial \mathbf{x}}=\mathbf{x}^T(\mathbf{A}+\mathbf{A}^T) \]
   • 若\(\mathbf A\)对称，则：\(\frac{\partial (\mathbf{x}^{T}\mathbf{A}\mathbf{x})}{\partial \mathbf{x}}=2\mathbf x^T\mathbf A\)
   • 结合性质2，还可以知道\(\frac{\partial(\mathbf{x}^{T}\mathbf{x})}{\partial \mathbf{z}}=2\mathbf{x}^T\frac{\partial\mathbf x}{\partial\mathbf z}\)

4. 已知\(\mathbf{x},\mathbf{y}\)是\(\mathbf{z}\)的函数，则：

   \[\frac{\partial (\mathbf y^T\mathbf A\mathbf x)}{\partial \mathbf{z}}=\mathbf{x}^{T}\mathbf A^T \frac{\partial \mathbf{y}}{\partial \mathbf{z}}+\mathbf{y}^{T}\mathbf A\frac{\partial \mathbf{x}}{\partial \mathbf{z}} \]

矩阵微分

矩阵对标量求微分的定义：

\[\frac{\partial \mathbf{A}}{\partial \alpha}=\left[\begin{array}{cccc} \frac{\partial a_{11}}{\partial \alpha} & \frac{\partial a_{12}}{\partial \alpha} & \cdots & \frac{\partial a_{1 \mathrm{n}}}{\partial \alpha} \\ \frac{\partial a_{21}}{\partial \alpha} & \frac{\partial a_{22}}{\partial \alpha} & \cdots & \frac{\partial a_{2 n}}{\partial \alpha} \\ \vdots & \vdots & & \vdots \\ \frac{\partial a_{\mathrm{m} 1}}{\partial \alpha} & \frac{\partial a_{\mathrm{m} 2}}{\partial \alpha} & \cdots & \frac{\partial a_{m n}}{\partial \alpha} \end{array}\right] \]

已知\(\mathbf{A}\)是\(\alpha\)的函数，则：

\[\frac{\partial \mathbf{A}^{-1}}{\partial \alpha}=-\mathbf{A}^{-1} \frac{\partial \mathbf{A}}{\partial \alpha} \mathbf{A}^{-1} \]

标签：常用,right,mathbf,运算,矩阵,alpha,frac,partial,21
From： https://www.cnblogs.com/harold-lu/p/16649582.html