数学杂记

数学

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任意一个方阵都可以表示成一个对称矩阵和一个反对称矩阵之和

对称矩阵：$W_{ij}=W_{ij}$，反对称矩阵：$W_{ij}=-W_{ji}$
现有方阵$A$

$$\begin{align*} B & = (A+A')/2 \\ C & = (A-A')/2 \\ A & = B+C \end{align*}$$
B为对称阵
$$\begin{align*} B_{ij} & = (A_{ij} + A_{ji})/2 \\ B_{ji} & = (A_{ji} + A_{ij})/2 = B_{ij} \end{align*}$$
C为反对称阵
$$\begin{align*} C_{ij} & = (A_{ij} - A_{ji})/2 \\ C_{ji} & = (A_{ji} - A_{ji})/2 = -C_{ji} \end{align*}$$

多项式分布中的分组公式

二项式分布是伯努力实验的计数，每次实验只有 $\{0,1\}$ 两种结果， $p(X=1)=\mu$ ，则在$N$次实验中出现$x=1$的次数符合二项式分布，

$$Bin(m|N,\mu)=\binom{N}{m}\mu^m(1-\mu)^{N-m}$$

$\binom{N}{m} = \frac{N!}{(N-m)!\cdot m!}$ 为组合数

多项式分布是二项式分布的扩展，每次实验可以有$k$种情况，则 $X=(x_1,x_2,\cdots,x_k)^T$，另一种说法是每一次实验的状态$X$服从一个离散分布$X \sim Discrete(\mu_1,\mu_2,\cdots,\mu_k)$，$\mu$表示相应状态出现的概率，是一个向量。

多项式分布也是一个关于计数的分布，表示的是在$N$次实验中每一种状态出现的次数的联合分布

$$Mult(m_1,m_2,\cdots,m_K|\mu,N)=\binom{N}{m_1 m_2 \cdots m_K}\prod_{k=1}^{K}\mu_k^{m_k}$$

系数也是一个组合数，意思是把$N$个物体分成$k$个不同的组，每个组的个数是 $(m_1,m_2,\cdots,m_K)$，且 $\sum_{k=1}^{K}m_k=N$

$$\begin{align*} \binom{N}{m_1 m_2 \cdots m_K} &= \binom{N}{m1} \cdot \binom{N-m_1}{m_2}\cdots \binom{N-m_1-m_2-\cdots-m_{K-1}}{m_K} \\ &=\frac{N!}{(N-m_1)!m_1!}\cdot\frac{(N-m_1)!}{(N-m_1-m_2)!m_2!}\cdots \frac{(N-m_1-m_2-\cdots-m_{K-1})}{(N-m_1-m_2-\cdots-m_{K-1}-m_K)!m_K!} \\ &=\frac{N!}{m_1!m_2!\cdots m_K!} \end{align*}$$

• 参考：从二项式分布到多项式分布-从Beta分布到Dirichlet分布

矩阵微分

分块矩阵的逆

$$A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \\ A^{-1} = \begin{bmatrix} (A_{11}-A_{12}A_{22}^{-1}A_{21})^{-1} & -A_{11}^{-1}A_{12}(A_{22}-A_{21}A_{11}^{-1}A_{12})^{-1} \\ -A_{22}^{-1}A_{21}(A_{11}-A_{12}A_{22}^{-1}A_{21})^{-1} & (A_{22}-A_{21}A_{11}^{-1}A_{12})^{-1} \end{bmatrix}$$

矩阵向量微分

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$$\begin{align*} \alpha & = y^TAx \\ \frac{\partial\alpha}{\partial x} & = y^TA \\ \frac{\partial\alpha}{\partial y} & = x^TA^T \end{align*}$$

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$$\begin{align*} \alpha & = x^TAx \\ \frac{\partial\alpha}{\partial x} & = x^T(A+A^T) \end{align*}$$

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$$\begin{align*} \alpha & = y^Tx \qquad //y(z),x(z) \\ \frac{\partial\alpha}{\partial z} & = x^T\frac{\partial y}{\partial z} + y^T\frac{\partial x}{\partial z} \\ \alpha & = x^Tx \\ \frac{\partial \alpha}{\partial z} & = 2x^T\frac{\partial x}{\partial z} \end{align*}$$

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$$\begin{align*} \alpha & = y^TAx \qquad //y(z),x(z)\\ \frac{\partial\alpha}{\partial z} & = x^TA^T\frac{\partial y}{\partial z} + y^TA\frac{\partial x}{\partial z} \end{align*}$$

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$$\frac{\partial A}{\partial \alpha} = \begin{bmatrix} \frac{\partial a_{11}}{\partial \alpha} & \frac{\partial a_{12}}{\partial \alpha} & \cdots & \frac{\partial a_{1n}}{\partial \alpha} \\ \frac{\partial a_{21}}{\partial \alpha} & \frac{\partial a_{22}}{\partial \alpha} & \cdots & \frac{\partial a_{2n}}{\partial \alpha} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial a_{m1}}{\partial \alpha} & \frac{\partial a_{m2}}{\partial \alpha} & \cdots & \frac{\partial a_{mn}}{\partial \alpha} \end{bmatrix} \\ \frac{\partial A^{-1}}{\partial \alpha} = -A^{-1}\frac{\partial A}{\partial \alpha}A^{-1}$$

反三角函数的导数

$$\begin{align*} y & = \arcsin x \\ x & = \sin y \\ 1 & = \cos y \frac{\mathrm{d}y}{\mathrm{d}x} \\ \frac{\mathrm{d}y}{\mathrm{d}x} & = \frac{1}{\cos y} \\ & = \frac{1}{\sqrt{1-\sin^2 y}} \\ & = \frac{1}{\sqrt{1-x^2}} \end{align*}$$