标签：Latex begin end color text frac cdots 符号表

Latex 公式速查

• Latex 公式速查
  • 函数
    • 对数与指数
    • 三角函数
    • 其他函数
  • 符号
    • 运算符
    • 集合
    • 关系符号
    • 几何符号
    • 逻辑符号
    • 箭头 - arrow
  • 希腊字母
  • 字体
    • 黑板报粗体
    • 粗体
    • 斜体
    • 无衬线体
    • 手写体
    • 注释文本
  • 颜色
  • 空格
  • 上下标与积分等
  • 分式
  • 矩阵
    • 无框矩阵 - matrix
    • 行列式 - vmatrix
    • 范数矩阵 - Vmatrix
    • 小括号矩阵 - pmatrix
    • 大括号矩阵 - Bmatrix
    • 方括号矩阵 - bmatrix
    • 边框 - boxed{}
  • 数组 - array
    • 定界符
      • 竖线
      • 小括号
      • 大括号
      • 方括号
    • 分割线
      • 实竖线
      • 虚竖线
      • 实横线 - \hline
      • 虚横线 - \hdashline
      • 应用 - 分块矩阵
      • 应用 - 制作表格
  • 条件表达式，方程式
    • 条件表达式 - cases
    • 编号的方程式 - equation
    • 多公式有编号 - align
    • 多公式无编号 - align*
      • 多公式无编号
      • 单方程式多行写
    • 自定义对齐方式
    • 方程组

本文仅提供的能够在 \(Markdown\) 中使用的 \(Latex\) 公式。

如何插入 \(Latex\) 公式？

• 行内公式：$公式$
• 独立公式：$$公式$$

函数

对数与指数

\(a^x\) a^x
\(\sqrt{x}\) \sqrt{x}
\(\sqrt[3]{x}\) \sqrt[3]{x}
\(\sqrt[a]{x}\) \sqrt[a]{x}
\(\exp x\) \exp x
\(\log x\) \log x
\(\lg x\) \lg x
\(\ln x\) \ln x

三角函数

\(\sin x\) \sin x
\(\cos x\) \cos x
\(\tan x\) \tan x
\(\cot x\) \cot x
\(\sec x\) \sec x
\(\csc x\) \csc x
\(\arcsin x\) \arcsin x
\(\arccos x\) \arccos x
\(\arctan x\) \arctan x
\(\sinh x\) \sinh x
\(\cosh x\) \cosh x
\(\tanh x\) \tanh x

其他函数

最小值： \(\min x\) \min x
最大值： \(\max x\) \max x
最大公约数： \(\gcd x\) \gcd x
角度： \(\deg\) \deg
极限： \(\lim_{x \to \infty}f(x)\) \lim_{x \to \infty}f(x)
上确界： \(\sup M\) \sup M
下确界： \(\inf M\) \inf M
行列式： \(\det A\) \det A
维数： \(\dim A\) \dim A
矩阵kernel： \(\ker A\) \ker A
投影： \(\Pr\) \Pr
同调群：\(\hom\) \hom
复数的幅角： \(\arg z\) \arg z
向下取整： \(\lfloor x \rfloor\) \lfloor x \rfloor
向上取整： \(\lceil x \rceil\) \lceil x \rceil
自定义函数： \(\operatorname{function} x\) \operatorname{function} x

符号

运算符

\(\pm\) \pm
\(\mp\) \mp
\(\dotplus\) \dotplus
\(\times\) \times
\(\div\) \div
\(\frac{a}{b}\) \frac{a}{b}
\(\divideontimes\) \divideontimes
\(\backslash\) \backslash
\(\cdot\) \cdot
\(\ast\) \ast
\(\circ\) \circ
\(\bullet\) \bullet
\(\boxplus\) \boxplus
\(\boxminus\) \boxminus
\(\boxtimes\) \boxtimes
\(\boxdot\) \boxdot
\(\oplus\) \oplus
\(\ominus\) \ominus
\(\otimes\) \otimes
\(\oslash\) \oslash
\(\odot\) \odot
\(\bigoplus\) bigoplus
\(\bigotimes\) \bigotimes
\(\bigodot\) \bigodot

集合

\(\{ \}\) \{ \}
\(\empty\) \empty
\(\varnothing\) \varnothing
\(\in\) \in
\(\not\in\) \notin 或 \not\in
\(\ni\) \ni
\(\notni\) \notni 或 \not\ni
\(\cap\) \cap
\(\Cap\) \Cap
\(\sqcap\) \sqcap
\(\bigcap\) \bigcap
\(\cup\) \cup
\(\Cup\) \Cup
\(\sqcup\) \sqcup
\(\bigcup\) \bigcup
\(\bigsqcup\) \bigscup
\(\uplus\) \uplus
\(\biguplus\) \biguplus
\(\subset\) \subset
\(\Subset\) \Subset
\(\sqsubset\) \sqsubset
\(\supset\) \supset
\(\Supset\) \Supset
\(\sqsupset\) \sqsupset
\(\subseteq\) \subseteq
\(\nsubseteq\) \nsubseteq
\(\subsetneq\) \subsetneq
\(\varsubsetneq\) \varsubsetneq
\(\sqsubseteq\) \sqsubseteq
\(\supseteq\) \supseteq
\(\nsupseteq\) \nsupseteq
\(\supsetneq\) \supsetneq
\(\varsupsetneq\) \varsupsetneq
\(\sqsupseteq\) \sqsupseteq
\(\sqsupset\) \sqsupset
\(\subseteqq\) \subseteqq
\(\nsubseteqq\) \nsubseteqq
\(\subsetneqq\) \subsetneqq
\(\varsubsetneqq\) \varsubsetneqq
\(\supseteqq\) \supseteqq
\(\nsupseteqq\) \nsupseteqq
\(\supsetneqq\) \supsetneqq
\(\varsupsetneqq\) \varsupsetneqq

关系符号

\(\ne\) \ne 或 \neq
\(\equiv\) \equiv
\(\not\equiv\) \not\equiv
\(\doteq\) \doteq
\(\doteqdot\) \doteqdot
\(\sim\) \sim
\(\nsim\) \nsim
\(\backsim\) \backsim
\(\thicksim\) \thicksim
\(\simeq\) \simeq
\(\backsimeq\) \backsimeq
\(\eqsim\) \eqsim
\(\cong\) \cong
\(\ncong\) \ncong
\(\approx\) \approx
\(\thickapprox\) \thickapprox
\(\approxeq\) \approxeq
\(\asymp\) \asymp
\(\propto\) \propto
\(\varpropto\) \varpropto
\(\ngtr\) \ngtr
\(\gg\) \gg
\(\ggg\) \ggg
\(\not\ggg\) \not\ggg
\(\gtrdot\) \gtrdot
\(\ngtr\) \ngtr
\(\lneq\) \lneq
\(\leqq\) \leqq
\(\nleq\) \nleq
\(\nleqq\) \nleqq
\(\lneqq\) \lneqq
\(\lvertneqq\) \lvertneqq
\(\ge\) \ge
\(\geq\) \geq
\(\gneq\) \gneq
\(\geqq\) \geqq
\(\ngeq\) \ngeq
\(\ngeqq\) \ngeqq
\(\gneqq\) \gneqq
\(\gvertneqq\) \gvertneqq

几何符号

\(\parallel\) \parallel
\(\nparallel\) \nparallel
\(\shortparallel\) \shortparallel
\(\nshortparallel\) nshortparallel
\(\perp\) \perp
\(\angle\) \angle
\(\sphericalangle\) \sphericalangle
\(\measuredangle\) \measuredangle
\(45^\circ\) 45^\circ
\(\Box\) \Box
\(\blacksquare\) \blacksquare
\(\diamond\) \diamond
\(\Diamond\) \Diamond
\(\lozenge\) \lozenge
\(\blacklozenge\) \blacklozenge
\(\bigstar\) \bigstar
\(\bigcirc\) \bigcirc
\(\triangle\) \triangle
\(\bigtriangleup\) \bigtriangleup
\(\bigtriangledown\) \bigtriangledown
\(\vartriangle\) \vartriangle
\(\triangledown\) \triangledown
\(\blacktriangle\) \blacktriangle
\(\blacktriangledown\) \blacktriangledown
\(\blacktriangleleft\) \blacktriangleleft
\(\blacktriangleright\) \blacktriangleright

逻辑符号

\(\forall\) \forall
\(\exists\) \exists
\(\nexists\) \nexists
\(\therefore\) \therefore
\(\because\) \because
\(\And\) \And
\(\mid\) \mid
\(\lor\) \lor 或 \vee
\(\land\) \land 或 \wedge
\(\bar{q}\) \bar{q}
\(\overline{q}\) \overline{q}
\(\lnot\) \lnot 或 \neg
\(\bot\) \bot
\(\top\) \top
\(\vdash\) \vdash
\(\dashv\) \dashv
\(\vDash\) \vDash
\(\Vdash\) \Vdash
\(\models\) \models
\(\ulcorner\) \ulcorner
\(\urcorner\) \urcorner
\(\llcorner\) \llcorner
\(\lrcorner\) \lrcorner

箭头 - arrow

\(\rightarrow\) \rightarrow
\(\nrightarrow\) \nrightarrow
\(\longrightarrow\) \longrightarrow
\(\Rightarrow\) \Rightarrow
\(\nRightarrow\) \nRightarrow
\(\Longrightarrow\) \Longrightarrow
\(\leftarrow\) \leftarrow
\(\nleftarrow\) n\leftarrow
\(\longleftarrow\) \longleftarrow
\(\Leftarrow\) \Leftarrow
\(\nLeftarrow\) \nLeftarrow
\(\Longleftarrow\) \Longleftarrow
\(\leftrightarrow\) \leftrightarrow
\(\nleftrightarrow\) \nleftrightarrow
\(\Leftrightarrow\) \Leftrightarrow
\(\nLeftrightarrow\) \nLeftrightarrow
\(\longleftrightarrow\) \longleftrightarrow
\(\iff\) iff
\(\Longleftrightarrow\) \Longleftrightarrow
\(\uparrow\) \uparrow
\(\downarrow\) \downarrow
\(\updownarrow\) \updownarrow
\(\Uparrow\) \Uparrow
\(\Downarrow\) \Downarrow
\(\nearrow\) \nearrow
\(\swarrow\) \swarrow
\(\nwarrow\) \nwarrow
\(\searrow\) \searrow
\(\rightharpoonup\) \rightharpoonup
\(\rightharpoondown\) \rightharpoondown
\(\leftharpoonup\) \leftharpoonup
\(\leftharpoondown\) \leftharpoondown
\(\upharpoonleft\) \upharpoonleft
\(\downharpoonleft\) \downharpoonleft
\(\upharpoonright\) \upharpoonright
\(\downharpoonright\) \downharpoonright
\(\rightleftharpoons\) \rightleftharpoons
\(\leftrightharpoons\) \leftrightharpoons
\(\curvearrowleft\) \curvearrowleft
\(\curvearrowright\) \curvearrowright
\(\circlearrowleft\) \circlearrowleft
\(\circlearrowright\) \circlearrowright
\(\Lsh\) \Lsh
\(\Rsh\) \Rsh
\(\upuparrows\) \upuparrows
\(\downdownarrows\) \downdownarrows
\(\leftleftarrows\) \leftleftarrows
\(\rightrightarrows\) \rightrightarrows
\(\stackrel{text}{\longrightarrow}\) \stackrel{text}{\longrightarrow}
\(\stackrel{text}{\longleftarrow}\) \stackrel{text}{\longleftarrow}
\(\stackrel{text}{\downarrow}\) \stackrel{text}{\downarrow}
\(\stackrel{text}{\uparrow}\) \stackrel{text}{\uparrow}

希腊字母

\(\alpha\) \alpha
\(\beta\) \beta
\(\gamma\) \gamma
\(\delta\) \delta
\(\epsilon\) \epsilon
\(\varepsilon\) \varepsilon
\(\zeta\) \zeta
\(\eta\) \eta
\(\theta\) \theta
\(\vartheta\) \vartheta
\(\iota\) \iota
\(\kappa\) \kappa
\(\lambda\) \lambda
\(\mu\) \mu
\(\nu\) \nu
\(\xi\) \xi
\(\pi\) \pi
\(\varpi\) \varpi
\(\rho\) \rho
\(\varrho\) \varrho
\(\sigma\) \sigma
\(\varsigma\) \varsigma
\(\tau\) \tau
\(\upsilon\) \upsilon
\(\phi\) \phi
\(\varphi\) \varphi
\(\chi\) \chi
\(\psi\) \psi
\(\omega\) \omega
\(\Gamma\) \Gamma
\(\Delta\) \Delta
\(\Theta\) \Theta
\(\Lambda\) \Lambda
\(\Xi\) \Xi
\(\Pi\) \Pi
\(\Sigma\) \Sigma
\(\Upsilon\) \Upsilon
\(\Phi\) \Phi
\(\Psi\) \Psi
\(\Omega\) \Omega

字体

黑板报粗体

只对大写字母有效

\(\mathbb{FONT}\) \mathbb{FONT}

粗体

对大小写字母、希腊字母都有效

\(\mathbf{FONT}\) \mathbf{FONT}
\(\mathbf{font}\) \mathbf{font}
\(\mathbf{\digamma\Theta\Nu\Tau}\) \mathbf{\digamma\Theta\Nu\Tau}

斜体

\(\mathit{1234567890}\) \mathit{1234567890}
\(\mathit{abcdefg}\) \mathit{abcdefg}
\(\mathit{ABCDEFG}\) \mathit{ABCDEFG}

无衬线体

\(\mathsf{ABCDEFG}\) \mathsf{ABCDEFG}

手写体

\(\mathcal{ABCDEFG}\) \mathcal{ABCDEFG}

注释文本

用 text{} 在公式中添加文本： \(\text{注释信息}\) \text{注释信息}

颜色

格式：

\color{颜色}{文本}

旧版浏览器支持：

\(\color{gray}{text}\) \color{gray}{text}
\(\color{silver}{text}\) \color{silver}{text}
\(\color{blue}{text}\) \color{blue}{text}
\(\color{yellow}{text}\) \color{yellow}{text}
\(\color{red}{text}\) \color{red}{text}
\(\color{lime}{text}\) \color{lime}{text}
\(\color{green}{text}\) \color{green}{text}
\(\color{fuchsia}{text}\) \color{fuchsia}{text}

较新浏览器支持 \color{#rgb}{text} 来自定义更多的颜色，#rgb 的 r、g、b 分别可以是十六进制表示的 0~255 的数。

\(\color{#ffdddd}{text}\) \color{#ffdddd}{text}
\(\color{#ff8888}{text}\) \color{#ff8888}{text}
\(\color{#ffaa11}{text}\) \color{#ffaa11}{text}
\(\color{#ffccaa}{text}\) \color{#ffccaa}{text}
\(\color{#ffdd66}{text}\) \color{#ffdd66}{text}
\(\color{#ffbbee}{text}\) \color{#ffbbee}{text}
\(\color{#aaaaff}{text}\) \color{#aaaaff}{text}
\(\color{#7777ff}{text}\) \color{#7777ff}{text}
\(\color{#66ccff}{text}\) \color{#66ccff}{text}
\(\color{#99ccff}{text}\) \color{#99ccff}{text}
\(\color{#00eeff}{text}\) \color{#00eeff}{text}
\(\color{#bbffee}{text}\) \color{#bbffee}{text}
\(\color{#99ff99}{text}\) \color{#99ff99}{text}
\(\color{#44bb66}{text}\) \color{#44bb66}{text}
\(\color{#44ff77}{text}\) \color{#44ff77}{text}
\(\color{#0088ff}{text}\) \color{#0088ff}{text}
\(\color{#22cc88}{text}\) \color{#22cc88}{text}
\(\color{#777777}{text}\) \color{#777777}{text}
\(\color{#aaaaaa}{text}\) \color{#aaaaaa}{text}
\(\color{#f0f0f0}{text}\) \color{#f0f0f0}{text}

空格

• \, 表示一个窄空格，\(\frac{1}{6}\) M 的宽度
• \ 或 \: 表示一个中等空格
• \; 表示一个大空格
• \quad 表示一个字母 M 宽度的空格
• \qquad 表示两个 \quad 的宽度
• \! 表示一个负的窄空格，缩进\(\frac{1}{6}M\) 的宽度
• \\ 表示换行

\[\boxed{ \begin{array}{c|c} 窄空格 & a\,b \\ \hline 中等空格 & a\: b \\ \hline 大空格 & a\;b \\ \hline 字母M的宽度 & a\quad b \\ \hline 两个M的宽度 & a\qquad b \\ \hline 负窄空格 & a\!b \end{array}\\ } \]

上下标与积分等

\(x^2\) x^2

\(x^{a + b}\) x^{a+b}

\(a_1\) a_1

\(a_{ij}\) a_{ij}

前置上下标： \({}_1^2\!X_3^4\) {}_1^2\!x_3^4

正上方标记：\(\sum\limits^n\) \sum\limits^n

正下方标记：\(\min\limits_{i \leq k \leq j - 1}\) \min\limits_{i \leq k \leq j - 1}

导数： \(x^\prime\) x^\prime 或 x'

导数点： \(\dot{x}\) \dot{x}

向量：\(\vec{x}\) \vec{x}

左长箭头： \(\overleftarrow{a + b}\) \overleftarrow{a + b}

右长箭头： \(\overrightarrow{a + b}\) \overrightarrow{a + b}

\(\widehat{abc}\) \widehate{abc}

上弧： \(\overset{\frown}{AB}\) \overset{\frown}{AB}

上划线： \(\overline{abc}\) \overline{abc}

下划线： \(\underline{abc}\) \underline{abc}

上括号： \(\overbrace{1 + 2 + \cdots + 100}\) \overbrace{1 + 2 + \cdots + 100}

上括号示例： \(\begin{matrix}5050\\\overbrace{1 + 2 + \cdots + 100}\end{matrix}\) \begin{matrix}5050\\\overbrace{1 + 2 + \cdots + 100}\end{matrix}

\(\overbrace{\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}}^{n\ \text{个根号}}\) \overbrace{\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}}^{n\ \text{个根号}}

下括号： \(\underbrace{1 + 2 + \cdots + 100}\) \underbrace{1 + 2 + \cdots + 100}

下括号示例： \(\begin{matrix}\underbrace{1 + 2 + \cdots + 100}\\5050\end{matrix}\) \begin{matrix}\underbrace{1 + 2 + \cdots + 100}\\5050\end{matrix}

\(\underbrace{\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}}_{n\ \text{个根号}}\) $\underbrace{\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}}_{n\ \text{个根号}}$

求和： \(\sum_{k = 1}^{\infty} f(x)\) \sum_{k = 1}^{\infty} f(x)

求和： \(\Sigma_{x = 1}^{\infty} f(x)\) \Sigma_{x = 1}^{t = \infty} f(x)

求积： \(\prod_{i = 1}^{n} x_i\) \prod_{i = 1}^{n} x_i

上积： \(\coprod_{i = 1}^{n} x_i\) \coprod_{i = 1}^{n} x_i

极限： \(\lim_{x\to\infty} f(x)\) \lim_{x\to\infty} f(x)

积分： \(\int_{a}^{b} f(x)dx\) \int_{a}^{b} f(x)dx

双重积分： \(\iint_{a}^{b} f(x) \, dx \, dy\) \iint_{a}^{b} f(x) \, dx \, dy

三重积分： \(\iiint_a^{b} f(x) \, dx \, dy \, dz\) \iiint_a^{b} f(x) \, dx \, dy \, dz

闭合的曲线、曲面积分： \(\oint_{C} x^2 \, dx+ y \, dy\) \oint_{C} x^2 \, dx+ y \, dy

分式

分数：
\(\frac{a + b}{c + d}\) \frac{a + b}{c + d}
\(\frac{dx}{dy}\) \frac{dx}{dy}

连分式： \(\cfrac{1}{2 + \cfrac{3}{4 + \cfrac{5}{6 + \cdots}}}\) \cfrac{1}{2 + \cfrac{3}{4 + \cfrac{5}{6 + \cdots}}}

\(\cfrac{a_1}{b1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}}}\) \cfrac{a_1}{b1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}}}

二项式系数： \(C_n^r = \dbinom{n}{r}\) C_n^r = \dbinom{n}{r}

矩阵

语法：

\begin{类型}
公式
\end{类型}

矩阵中 & 分隔元素，\\ 进行换行
横三点： \(\cdots\) \cdots
竖三点： \(\vdots\) \vdots
斜三点： \(\ddots\) \ddots

无框矩阵 - matrix

\[\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}\\ \]

\begin{matrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{matrix}

行列式 - vmatrix

\[\begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{21} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{vmatrix}\\ \]

\begin{vmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{21} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn}
\end{vmatrix}

范数矩阵 - Vmatrix

\[\begin{Vmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,1} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{Vmatrix}\\ \]

\begin{Vmatrix}
a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\
a_{2,1} & a_{2,1} & \cdots & a_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n,1} & a_{n,2} & \cdots & a_{n,n}
\end{Vmatrix}

小括号矩阵 - pmatrix

\[\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}\\ \]

\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}

大括号矩阵 - Bmatrix

\[\begin{Bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{Bmatrix}\\ \]

\begin{Bmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
a_{41} & a_{42} & a_{43} & a_{44}  
\end{Bmatrix}

方括号矩阵 - bmatrix

\[\begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn} \end{bmatrix}\\ \]

\begin{bmatrix}
a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\
a_{21} & a_{22} & a_{23} & \cdots & a_{2n} \\
a_{31} & a_{32} & a_{33} & \cdots & a_{3n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn} 
\end{bmatrix}

边框 - boxed{}

\[\begin{bmatrix} \boxed{-1} & 3 & 0 & 2 \\ 0 & \boxed{1} & 3 & 1 \\ 0 & 0 & 0 & \boxed{2} \\ 0 & 0 & 0 & 0 \end{bmatrix}\\ \]

\begin{bmatrix}
\boxed{-1} & 3 & 0 & 2 \\
0 & \boxed{1} & 3 & 1 \\
0 & 0 & 0 & \boxed{2} \\
0 & 0 & 0 & 0
\end{bmatrix}\\

数组 - array

\[\begin{array}{} a & b \\ c & d \end{array}\\ \]

\begin{array}{}
a & b \\
c & d  
\end{array}

定界符

语法：

\left 符号
公式
\right 符号

竖线

\[\left | \begin{array}{} a_{11} & a_{12} \\ a_{13} & a_{14} \\ \end{array} \right | \\ \]

\left |
\begin{array}{}
a_{11} & a_{12} \\
a_{13} & a_{14} \\
\end{array}
\right | 

小括号

\[\left ( \begin{array}{} a_{11} & a_{12} \\ a_{13} & a_{14} \\ \end{array} \right ) \\ \]

\left (
\begin{array}{}
a_{11} & a_{12} \\
a_{13} & a_{14} \\
\end{array}
\right )

大括号

\[\left \{ \begin{array}{} a_{11} & a_{12} \\ a_{13} & a_{14} \\ \end{array} \right \}\\ \]

\left \{
\begin{array}{}
a_{11} & a_{12} \\
a_{13} & a_{14} \\
\end{array}
\right \}

注：{} 为特殊字符，无法直接使用，应使用 \{ 和 \} 来输出

方括号

\[\left [ \begin{array}{} a_{11} & a_{12} \\ a_{13} & a_{14} \\ \end{array} \right ]\\ \]

\left [
\begin{array}{}
a_{11} & a_{12} \\
a_{13} & a_{14} \\
\end{array}
\right ]

分割线

实竖线

\[\left [ \begin{array}{c|c|c|c|c} a_{11} & a_{12} & a_{13} & a_{14} & a_{15}\\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25}\\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35}\\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45}\\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right ]\\ \]

\left [
\begin{array}{c|c|c|c|c}
a_{11} & a_{12} & a_{13} & a_{14} & a_{15}\\
a_{21} & a_{22} & a_{23} & a_{24} & a_{25}\\
a_{31} & a_{32} & a_{33} & a_{34} & a_{35}\\
a_{41} & a_{42} & a_{43} & a_{44} & a_{45}\\
a_{51} & a_{52} & a_{53} & a_{54} & a_{55} 
\end{array}\\
\right ]

虚竖线

\[\left [ \begin{array}{c:c:c:c:c} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right ]\\ \]

\left [
\begin{array}{c:c:c:c:c}
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\
a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\
a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\
a_{51} & a_{52} & a_{53} & a_{54} & a_{55}
\end{array}
\right ]\\

实横线 - \hline

\[\left [ \begin{array}{} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ \hline a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ \hline a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ \hline a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ \hline a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right ]\\ \]

\left [
\begin{array}{}
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
\hline
a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\
\hline
a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\
\hline
a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\
\hline
a_{51} & a_{52} & a_{53} & a_{54} & a_{55}
\end{array}
\right ]

虚横线 - \hdashline

\[\left [ \begin{array}{} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ \hdashline a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ \hdashline a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ \hdashline a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ \hdashline a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{array} \right ]\\ \]

\left [
\begin{array}{}
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
\hdashline
a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\
\hdashline
a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\
\hdashline
a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\
\hdashline
a_{51} & a_{52} & a_{53} & a_{54} & a_{55}
\end{array}
\right ]

应用 - 分块矩阵

\[\left [ \begin{array}{cc:cc} 1 & 0 & 1 & -2 \\ 0 & 1 & 0 & 1 \\ \hdashline -1 & 2 & -1 & 0 \\ 0 & -1 & 0 & -1 \end{array} \right ]\\ \]

\left [
\begin{array}{cc:cc}
1 & 0 & 1 & -2 \\
0 & 1 & 0 & 1 \\
\hdashline
-1 & 2 & -1 & 0 \\
0 & -1 & 0 & -1
\end{array}
\right ]

应用 - 制作表格

\[\boxed{ \begin{array}{c|c} 矩阵类型 & 关键字 \\ \hline |A| & vmatrix \\ \hline \parallel & Vmatrix \\ \hline () & pmatrix \\ \hline \{\} & Bmatrix \\ \hline [\ ] & bmatrix \end{array} }\\ \]

\boxed{
    \begin{array}{c|c}
    矩阵类型 & 关键字 \\ \hline
    |A| & vmatrix \\ \hline
    \parallel & Vmatrix \\ \hline
    () & pmatrix \\ \hline
    \{\} & Bmatrix \\ \hline
    [\ ] & bmatrix
    \end{array}
}

条件表达式，方程式

条件表达式 - cases

\[f(x) = \begin{cases} \begin{aligned} \frac{\sin x}{|x|},x \ne 0 \\ 1,x = 0\\ \end{aligned} \end{cases}\\ \]

f(x) =
\begin{cases}
\begin{aligned}
\frac{\sin x}{|x|},x \ne 0 \\
1,x = 0\\
\end{aligned}
\end{cases}

编号的方程式 - equation

\[\begin{equation} z = (a+b)^4= a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4. \end{equation}\\ \]

\begin{equation}
z = (a+b)^4= a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.
\end{equation}

多公式有编号 - align

\[\begin{align} \nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} \\ \nabla \cdot \mathbf{B} &= 0 \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{align}\\ \]

\begin{align} 
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} \\
 \nabla \cdot \mathbf{B} &= 0 \\
 \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\
 \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
 \end{align}

多公式无编号 - align*

多公式无编号

\[\begin{align*} E = mc^2 \\ e^{i\pi} + 1 = 0 \end{align*}\\ \]

\begin{align*}
E = mc^2 \\
e^{i\pi} + 1 = 0
\end{align*}

单方程式多行写

\[\begin{align*} z & = (a+b)^4 \\ & = (a+b)^2(a+b)^2 \\ & = (a^2+2ab+b^2)(a^2+2ab+b^2) \\ & = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 \end{align*}\\ \]

\begin{align*}
   z & = (a+b)^4 \\
     & = (a+b)^2(a+b)^2 \\
     & = (a^2+2ab+b^2)(a^2+2ab+b^2) \\
     & = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
 \end{align*}

\[\begin{align*} & a_1 \wedge a_2 \wedge \cdots \wedge (a_i \wedge x) \wedge a_{i + 1} \wedge \cdots \wedge a_n\\ & = (a_1 \wedge a_2 \wedge \cdots \wedge a_i \wedge a_{i + 1} \wedge \cdots \wedge a_n) \wedge x\\ & = x \wedge x\\ & = 0 \end{align*} \]

\begin{align*}
& a_1 \wedge a_2 \wedge \cdots \wedge (a_i \wedge x) \wedge a_{i + 1} \wedge \cdots \wedge a_n\\
& = (a_1 \wedge a_2 \wedge \cdots \wedge a_i \wedge a_{i + 1} \wedge \cdots \wedge a_n) \wedge x\\
& = x \wedge x\\
& = 0
\end{align*}

自定义对齐方式

在 align 或 align* 环境下，在公式左侧添加 & 可以使得公式左对齐，否则默认为居中对齐。
实际上将 & 的作用是为公式设置一个对齐点，多个公式的对齐点会在同一竖线上。

• 将标记的 \(=\) 和 \(+\) 之间保持对齐，在对应 = 和 + 前标记 &

\[\begin{align*} \phi(N) &= N - \frac{N}{p_1} - \frac{N}{p_2} \cdots - \frac{N}{p_k}\\ & +\frac{N}{p_1p_2} + \frac{N}{p_1p_3} + \cdots\\ & +\frac{N}{p_1p_2p_3} - \frac{N}{p_2p_2p_4} \cdots\\ & +\cdots \\ & = N \times \frac{p_1 - 1}{p_1} \times \frac{p_2 - 1}{p_2}\cdots \times \frac{p_m - 1}{p_m} \end{align*} \]

\begin{align*}
\phi(N) &= N - \frac{N}{p_1} - \frac{N}{p_2} \cdots - \frac{N}{p_k}\\
& +\frac{N}{p_1p_2} + \frac{N}{p_1p_3} + \cdots\\
& +\frac{N}{p_1p_2p_3} - \frac{N}{p_2p_2p_4} \cdots\\
& +\cdots \\
& = N \times \frac{p_1 - 1}{p_1} \times \frac{p_2 - 1}{p_2}\cdots \times \frac{p_m - 1}{p_m}
\end{align*}

• 将 \(\cdots\) 之间保持对齐，在所有 \cdots 前标记 &

\[\begin{align*} \phi(N) = N - \frac{N}{p_1} - \frac{N}{p_2} &\cdots - \frac{N}{p_k}\\ +\frac{N}{p_1p_2} + \frac{N}{p_1p_3} + &\cdots\\ +\frac{N}{p_1p_2p_3} - \frac{N}{p_2p_2p_4} &\cdots\\ +&\cdots \\ = N \times \frac{p_1 - 1}{p_1} \times \frac{p_2 - 1}{p_2} &\cdots \times \frac{p_m - 1}{p_m} \end{align*} \]

\begin{align*}
\phi(N) = N - \frac{N}{p_1} - \frac{N}{p_2} &\cdots - \frac{N}{p_k}\\
+\frac{N}{p_1p_2} + \frac{N}{p_1p_3} + &\cdots\\
+\frac{N}{p_1p_2p_3} - \frac{N}{p_2p_2p_4} &\cdots\\
+ &\cdots \\
= N \times \frac{p_1 - 1}{p_1} \times \frac{p_2 - 1}{p_2} &\cdots \times \frac{p_m - 1}{p_m}
\end{align*}

方程组

\[\begin{cases} x + y - z = 0 \\ 2x - y + z = 2 \\ x + y + 2z = 4 \end{cases}\\ \]

\begin{cases}
x + y - z = 0 \\
2x - y + z = 2 \\
x + y + 2z = 4
\end{cases}

或者

\left\{ \begin{aligned}
x + y - z = 0 \\
2x - y + z = 2 \\
x + y + 2z = 4
\end{aligned} \right.

\left\{ 公式 \right. 实现只有左边出现界定符大括号 {
\begin{aligned} 公式 \end{aligned} 实现公式右对齐

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标签：Latex,begin,end,color,text,frac,cdots,符号表
From： https://www.cnblogs.com/YzaCsp/p/18574398