Latex常用内容

\[X=\left| \begin{matrix} x_{11} & x_{12} & \cdots & x_{1d}\\ x_{21} & x_{22} & \cdots & x_{2d}\\ \vdots & \vdots & \ddots & \vdots \\ x_{11} & x_{12} & \cdots & x_{1d}\\ \end{matrix} \right| \]

\[\begin{matrix} 1 & x & x^2\\ 1 & y & y^2\\ 1 & z & z^2\\ \end{matrix} \]

\[\left\{ \begin{array}{c} a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2\\ a_3x+b_3y+c_3z=d_3 \end{array} \right\} \]

\[X=\begin{pmatrix} 0&1&1\\ 1&1&0\\ 1&0&1\\ \end{pmatrix} \]

1. 希腊字母表

\Sigma: \(\Sigma\)

\[\begin{align*} RQSZ \\ \mathcal{RQSZ} \\ \mathfrak{RQSZ} \\ \mathbb{RQSZ} \end{align*} \]

\[\begin{align*} 3x^2 \in R \subset Q \\ \mathnormal{3x^2 \in R \subset Q} \\ \mathrm{3x^2 \in R \subset Q} \\ \mathit{3x^2 \in R \subset Q} \\ \mathbf{3x^2 \in R \subset Q} \\ \mathsf{3x^2 \in R \subset Q} \\ \mathtt{3x^2 \in R \subset Q} \end{align*} \]

2. 上下标、根号、省略号、空格

• 下标：_ x^2 \(\Longrightarrow\) $ x^2$

• 上标：^ x_i\(\Longrightarrow\) \(x_i\)

• 根号：\sqrt | y\sqrt{x}\(\Longrightarrow\) \(y\sqrt{x}\)

• 省略号：

  \dots \(\Longrightarrow\dots\)

  \cdots \(\Longrightarrow\cdots\)

  \ddots \(\Longrightarrow\ddots\)

• 括号

两个quad空格	a \qquad b	\(a \qquad b\)	两个m的宽度
quad空格	a \quad b	\(a \quad b\)	一个m的宽度
大空格	a\ b	\(a\ b\)	1/3m宽度
中等空格	a;b	\(a\;b\)	2/7m宽度
小空格	a,b	\(a\,b\)	1/6m宽度
没有空格	ab	\(ab\)
紧贴	a!b	\(a\!b\)	缩进1/6m宽度

3. 运算符

• 求和： \sum_1^n\(\Longrightarrow\) \(\sum_1^n\)
• 积分：\int_1^n \(\Longrightarrow\) \(\int_1^n\)
• 极限：lim_{x \to \infty}\(\Longrightarrow\) \(lim_{x \to \infty}\)
• 分数：\frac{2}{3} \(\Longrightarrow\) $\frac{2}{3} $
• 开方：\sqrt[2]{x} \(\Longrightarrow\) \(\sqrt[2]{x}\)
• 积 ：\prod_{i=0}^n \(\Longrightarrow\) \(\prod_{i=0}^n\)

f4. 箭头

字符	含义
\uparrow	↑
\downarrow	↓
\Uparrow	⇑
\Downarrow	⇓
\updownarrow	↕
\Updownarrow	⇕
\rightarrow	→
\leftarrow	←
\Rightarrow	⇒
\Leftarrow	⇐
\leftrightarrow	↔
\Leftrightarrow	⇔
\longrightarrow	⟶
\longleftarrow	⟵
\Longrightarrow	⟹
\Longleftarrow	⟸
\mapsto	↦
\longmapsto	⟼
\hookleftarrow	↩
\hookrightarrow	↪
\leftharpoonup	↼
\rightharpoonup	⇀
\leftharpoondown	↽
\rightharpoondown	⇁
\rightleftharpoons	⇌
\leadsto	⇝
\nearrow	↗
\searrow	↘
\swarrow	↙
\nwarrow	↖
\nleftarrow	↚
\nrightarrow	↛
\nLeftarrow	⇍
\nRightarrow	⇏
\nleftrightarrow	↮
\nLeftrightarrow	⇎
\dashrightarrow	⇢
\dashleftarrow	⇠
\leftleftarrows	⇇
\leftrightarrows	⇆
\Lleftarrow	⇚
\twoheadleftarrow	↞
\leftarrowtail	↢
\looparrowleft	↫
\leftrightharpoons	⇋
\curvearrowleft	↶
\circlearrowleft	↺
\Lsh	↰
\upharpoonleft	↿
\downharpoonleft	⇃
\upuparrows	⇈
\multimap	⊸
\leftrightsquigarrow	↭
\rightrightarrows	⇉
\rightleftarrows	⇄
\twoheadrightarrow	↠
\rightarrowtail	↣
\looparrowright	↬
\rightleftharpoons	⇌
\curvearrowright	↷
\circlearrowright	↻
\Rsh	↱
\downdownarrows	⇊
\upharpoonright	↾
\downharpoonright	⇂
\rightsquigarrow	⇝

5. 分段函数

f(n)=
	\begin{cases}
		n/2, & \text{if $n$ is even}\\
		3n+1,& \text{if $n$ is odd}
	\end{cases}

\[f(n)= \begin{cases} n/2, & \text{if $n$ is even}\\ 3n+1,& \text{if $n$ is odd} \end{cases} \]

6. 方程组

\left.
  \left\{
    \begin{array}{c}
      a_1x+b_1y+c_1z=d_1\\
      a_2x+b_2y+c_2z=d_2\\
      a_3x+b_3y+c_3z=d_3
    \end{array}
  \right.
  \right>

\[ \left. \left\{ \begin{array}{c} a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2\\ a_3x+b_3y+c_3z=d_3 \end{array} \right. \right> \]

7.矩阵

7.1 基本语法

• 起始标记 \begin{matrix}，结束标记 \end{matrix}
• 每一行末尾标记 \\
• 行间元素之间用 & 分隔。

\begin{matrix}
0&1&1\\
1&1&0\\
1&0&1\\
\end{matrix}

\[ \begin{matrix} 0&1&1\\ 1&1&0\\ 1&0&1\\ \end{matrix} \]

7.2 矩阵边框

• 在起始、结束标记用下列词替换 matrix
• pmatrix：小括号边框
• bmatrix：中括号边框
• Bmatrix：大括号边框
• vmatrix：单竖线边框
• Vmatrix：双竖线边框

\begin{vmatrix}
0&1&1\\
1&1&0\\
1&0&1\\
\end{vmatrix}

\[\begin{vmatrix} 0&1&1\\ 1&1&0\\ 1&0&1\\ \end{vmatrix} \]

7.3 省略元素

• 横省略号：\cdots
• 竖省略号：\vdots
• 斜省略号：\ddots

\begin{bmatrix}
{a_{11}}&{a_{12}}&{\cdots}&{a_{1n}}\\
{a_{21}}&{a_{22}}&{\cdots}&{a_{2n}}\\
{\vdots}&{\vdots}&{\ddots}&{\vdots}\\
{a_{m1}}&{a_{m2}}&{\cdots}&{a_{mn}}\\
\end{bmatrix}

\[\begin{bmatrix} {a_{11}}&{a_{12}}&{\cdots}&{a_{1n}}\\ {a_{21}}&{a_{22}}&{\cdots}&{a_{2n}}\\ {\vdots}&{\vdots}&{\ddots}&{\vdots}\\ {a_{m1}}&{a_{m2}}&{\cdots}&{a_{mn}}\\ \end{bmatrix} \]

7.4 阵列

• 需要array环境：起始、结束处以{array}声明
• 对齐方式：在{array}后以{}逐行统一声明
• 左对齐：l 居中:c 右对齐：r
• 竖直线：在声明对齐方式时，插入 | 建立竖直线
• 插入水平线：\hline

\begin{array}{c|lll}
{↓}&{a}&{b}&{c}\\
\hline
{R_1}&{c}&{b}&{a}\\
{R_2}&{b}&{c}&{c}\\
\end{array}

\[\begin{array}{c|lll} {↓}&{a}&{b}&{c}\\ \hline {R_1}&{c}&{b}&{a}\\ {R_2}&{b}&{c}&{c}\\ \end{array} \]

8.常用公式

8.1 线性模型

h(\theta) = \sum_{j=0} ^n \theta_j x_j

\[h(\theta) = \sum_{j=0} ^n \theta_j x_j \]

8.2 均方误差

J(\theta) = \frac{1}{2m}\sum_{i=0}^m(y^i - h_\theta(x^i))^2

\[J(\theta) = \frac{1}{2m}\sum_{i=0}^m(y^i - h_\theta(x^i))^2 \]

8.3 求积

H_c=\sum_{l_1+\dots +l_p}\prod^p_{i=1} \binom{n_i}{l_i}

\[H_c=\sum_{l_1+\dots +l_p}\prod^p_{i=1} \binom{n_i}{l_i} \]

8.4 批梯度下降

\[\frac{\partial J(\theta)}{\partial\theta_j} = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i))x^i_j \]

\[\begin{align} \frac{\partial J(\theta)}{\partial\theta_j} & = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(y^i-h_\theta(x^i))\\ & = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(\sum_{j=0}^n\theta_j x^i_j-y^i)\\ &=-\frac1m\sum_{i=0}^m(y^i -h_\theta(x^i)) x^i_j \end{align} \]

---

8.5 贝叶斯、先验、后验估计、似然估计

\[\begin{align} \overbrace{P(x,y|z,u)}^{后验概率} &= \frac{P(z,u|x,y)P(x,y)}{P(z,u)}\\ &\approx \underbrace{P(z,u|x,y)}_{似然} \ \underbrace{P(x,y)}_{先验概率} \end{align} \]

\[\begin{align} (x,y)^*_{MAP} &=argmax{P(x,y∣z,u)} \\ &=argmax{\frac{P(z,u|x,y)P(x,y)}{\underbrace{P(z,u)}_{此项与x,y无关，可以去掉}}} \\ &=argmax{{P(z,u|x,y)P(x,y)}} \end{align} \]

8.6 任意维高斯分布

\[P(x)=\frac{1}{\sqrt{(2\pi)^Ndet(\Sigma)}}exp(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)) \\ \Downarrow\Downarrow\Downarrow 负对数\\ -ln(P(x)) = \frac{1}{2}ln((2\pi)^Ndet(\Sigma))+\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu) \\ \Downarrow\Downarrow\Downarrow 负对数\\ (x)^*_{MLE}=argmax(P(x))=argmin((x-\mu)^T\Sigma^{-1}(x-\mu)) \]

8.7 高斯牛顿法求导

\[\begin{align} \triangle x^* &= \underset{\triangle x}{argmin} \frac{1}{2}||f(x+\triangle x)||^2 \\ &\approx ||f(x) + J(x)^T \triangle x ||^2 \\ &=\frac{1}{2}({||f(x)||^2+2f(x)J(x)^T\triangle x + \triangle x^TJ(x)J(x)^T\triangle x}) \\ \\ &\downdownarrows{令其求导等于0} \\ \\ &\underbrace{J(x)J(x)^T}_{H(x)}\triangle x = \underbrace{-J(x)f(x) }_{g(x)} \end{align} \\ \]

引用：

• Typora中利用LaTeX 插入数学公式
• 如何在 markdown 中表示矩阵？