Latex学习
\[X=\left|
\begin{matrix}
x_{11} & x_{12} & \cdots & x_{1d}\\
x_{21} & x_{22} & \cdots & x_{2d}\\
\vdots & \vdots & \ddots & \vdots \\
x_{11} & x_{12} & \cdots & x_{1d}\\
\end{matrix}
\right|
\]
\[\begin{matrix}
1 & x & x^2\\
1 & y & y^2\\
1 & z & z^2\\
\end{matrix}
\]
\[\left\{
\begin{array}{c}
a_1x+b_1y+c_1z=d_1\\
a_2x+b_2y+c_2z=d_2\\
a_3x+b_3y+c_3z=d_3
\end{array}
\right\}
\]
\[X=\begin{pmatrix}
0&1&1\\
1&1&0\\
1&0&1\\
\end{pmatrix}
\]
1. 希腊字母表
\Sigma: \(\Sigma\)
2. 上下标、根号、省略号
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
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} $
4. 箭头
\leftarrow对应 \(\leftarrow\)
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
matrixpmatrix：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}
\[\begin{array}{c|lll}
{↓}&{a}&{b}&{c}\\
\hline
{R_1}&{c}&{b}&{a}\\
{R_2}&{b}&{c}&{c}\\
\end{array}
\]
- 需要array环境：起始、结束处以{array}声明
7.5 等号上下文字
\arrowname[sub-script]{super-script}
- arrowname具体见下面，等号名称
- sub-script 代表等号下面内容
- super-script 代表等号上面内容
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
\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}
\[\frac{\partial J(\theta)}{\partial\theta_j} = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i))x^i_j
\]
\[\begin{align}
\frac{\partial J(\theta)}{\partial\theta_j}
& = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(y^i-h_\theta(x^i))\\
& = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(\sum_{j=0}^n\theta_j x^i_j-y^i)\\
&=-\frac1m\sum_{i=0}^m(y^i -h_\theta(x^i)) x^i_j
\end{align}
\]
