514 lines
22 KiB
HTML
514 lines
22 KiB
HTML
|
||
<!DOCTYPE html>
|
||
<html lang="zh-CN" class="loading">
|
||
<head>
|
||
<!-- hexo-inject:begin --><!-- hexo-inject:end --><meta charset="UTF-8" />
|
||
<meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1" />
|
||
<meta name="viewport" content="width=device-width, minimum-scale=1.0, maximum-scale=1.0, user-scalable=no">
|
||
<title>想法速记：SVD求齐次线性方程组 Ax=0 的解 - Blog</title>
|
||
<meta name="apple-mobile-web-app-capable" content="yes" />
|
||
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
|
||
<meta name="google" content="notranslate" />
|
||
<meta name="keywords" content="Developer, Python, C++, JavaScript, Java,">
|
||
<meta name="description" content="DavidZ's Blog,
|
||
原创发表于 DavidZ Blog，遵循 CC 4.0 BY-NC-SA 版权协议，转载请附上原文出处链接及本声明。
|
||
|
||
众所周知，SVD(奇异值分解)可以用于最小二乘法求齐次线性方程组$A\vec,">
|
||
<meta name="author" content="DavidZ">
|
||
<link rel="alternative" href="atom.xml" title="Blog" type="application/atom+xml">
|
||
<link rel="icon" href="/img/favicon.png">
|
||
|
||
<link rel="stylesheet" href="//at.alicdn.com/t/font_1429596_nzgqgvnmkjb.css">
|
||
|
||
|
||
<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/animate.css@3.7.2/animate.min.css">
|
||
|
||
|
||
<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/social-share.js@1.0.16/dist/css/share.min.css">
|
||
|
||
|
||
<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/codemirror@5.48.4/lib/codemirror.min.css">
|
||
|
||
|
||
<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/codemirror@5.48.4/theme/dracula.css">
|
||
|
||
|
||
<link rel="stylesheet" href="/css/obsidian.css">
|
||
|
||
|
||
<link rel="stylesheet" href="/css/ball-atom.min.css">
|
||
|
||
|
||
|
||
<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/font-awesome/css/font-awesome.min.css">
|
||
|
||
<meta name="generator" content="Hexo 4.2.1"><!-- hexo-inject:begin --><!-- hexo-inject:end --></head>
|
||
|
||
|
||
<body class="loading">
|
||
<!-- hexo-inject:begin --><!-- hexo-inject:end --><div class="loader">
|
||
<div class="la-ball-atom la-2x">
|
||
<div></div>
|
||
<div></div>
|
||
<div></div>
|
||
<div></div>
|
||
</div>
|
||
</div>
|
||
<span id="config-title" style="display:none">Blog</span>
|
||
<div id="loader"></div>
|
||
<div id="single">
|
||
<div class="scrollbar gradient-bg-rev"></div>
|
||
<div id="top" style="display: block;">
|
||
<div class="bar" style="width: 0;"></div>
|
||
<div class="navigation animated fadeIn fast delay-1s">
|
||
<img id="home-icon" class="icon-home" src="/img/favicon.png" alt="" data-url="https://blog.davidz.cn">
|
||
<div id="play-icon" title="Play/Pause" class="iconfont icon-play"></div>
|
||
<h3 class="subtitle">想法速记：SVD求齐次线性方程组 Ax=0 的解</h3>
|
||
<div class="social">
|
||
<!-- <div class="like-icon">-->
|
||
<!-- <a href="javascript:;" class="likeThis active"><span class="icon-like"></span><span class="count">76</span></a>-->
|
||
<!-- </div>-->
|
||
<div>
|
||
<div class="share">
|
||
|
||
<a href="javascript:;" class="iconfont icon-share1"></a>
|
||
<div class="share-component-cc" data-disabled="facebook,douban,linkedin,diandian,tencent,google"></div>
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div class="section">
|
||
<div class=article-header-wrapper>
|
||
<div class="article-header">
|
||
<div class="article-cover animated fadeIn" style="
|
||
animation-delay: 600ms;
|
||
animation-duration: 1.2s;
|
||
background-image:
|
||
radial-gradient(ellipse closest-side, rgba(0, 0, 0, 0.65), #100e17),
|
||
url(https://davidz-blog.oss-cn-beijing.aliyuncs.com/img/idea-1602143497.jpg) ">
|
||
</div>
|
||
<div class="else">
|
||
<p class="animated fadeInDown">
|
||
|
||
<a href="/categories/Idea"><b>「
|
||
</b>IDEA<b> 」</b></a>
|
||
|
||
六月 04, 2020
|
||
</p>
|
||
<h3 class="post-title animated fadeInDown"><a href="/SVD-for-homogeneous-linear-equation/" title="想法速记：SVD求齐次线性方程组 Ax=0 的解" class="">想法速记：SVD求齐次线性方程组 Ax=0 的解</a>
|
||
</h3>
|
||
|
||
<p class="post-count animated fadeInDown">
|
||
|
||
<span>
|
||
<b class="iconfont icon-text2"></b> <i>文章字数</i>
|
||
2.9k
|
||
</span>
|
||
|
||
|
||
<span>
|
||
<b class="iconfont icon-timer__s"></b> <i>阅读约需</i>
|
||
3 mins.
|
||
</span>
|
||
|
||
|
||
|
||
<span id="busuanzi_container_page_pv">
|
||
<b class="iconfont icon-read"></b> <i>阅读次数</i>
|
||
<span id="busuanzi_value_page_pv">0</span>
|
||
</span>
|
||
|
||
</p>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div class="screen-gradient-after">
|
||
<div class="screen-gradient-content">
|
||
<div class="screen-gradient-content-inside">
|
||
<div class="bold-underline-links screen-gradient-sponsor">
|
||
<p>
|
||
<span class="animated fadeIn delay-1s"></span>
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div class="article">
|
||
<div class='main'>
|
||
<div class="content markdown animated fadeIn">
|
||
<blockquote>
|
||
<p>原创发表于 <a href="https://blog.davidz.cn">DavidZ Blog</a>，遵循 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode" target="_blank" rel="noopener">CC 4.0 BY-NC-SA</a> 版权协议，转载请附上原文出处链接及本声明。</p>
|
||
</blockquote>
|
||
<p>众所周知，SVD(奇异值分解)可以用于最小二乘法求齐次线性方程组$A\vec{x}=\vec{0}$的解。我看了很多资料，大多使用数学公式推导，得出结论。但是，曾经线性代数差点挂科的我，总觉得有些蹊跷。想了两天，终于有了一些感性的认知（不一定是对的😂），赶紧记录下来。</p>
|
||
<h2 id="矩阵有何意义"><a class="header-anchor" href="#矩阵有何意义">¶</a>矩阵有何意义</h2>
|
||
<p>按照我的理解，一个矩阵的实际意义是对应一个线性变换，这个变换可以理解为瞬间运动。例如，一个旋转矩阵，</p>
|
||
<p>$$<br>
|
||
A=\begin{bmatrix}<br>
|
||
cos\theta & sin\theta \\\<br>
|
||
-sin\theta & cos\theta<br>
|
||
\end{bmatrix}<br>
|
||
$$</p>
|
||
<p>它的意思是，把一个向量顺时针旋转$\theta$。也就是说，给定一个$\vec{v_1}=[-1, 1]^T$, 那么变换的结果就是$\vec{v_2}=A\vec{v_1}=[1, 1]^T$.</p>
|
||
<p>除了旋转，矩阵还可以表示包括缩放，投影在内的所有<strong>线性变换</strong>。</p>
|
||
<p>十分推荐大家去看 3Blue1Brown 的 <em>线性代数的本质</em>，B 站有<a href="https://www.bilibili.com/video/BV1ys411472E" target="_blank" rel="noopener">官方翻译版</a>，它完全颠覆了我对线性代数的认知。</p>
|
||
<h2 id="SVD到底干了什么"><a class="header-anchor" href="#SVD到底干了什么">¶</a>SVD到底干了什么</h2>
|
||
<p>$$<br>
|
||
A = U\Sigma V^T<br>
|
||
$$</p>
|
||
<p>SVD把矩阵$A(m\times n)$分解成了,</p>
|
||
<ul>
|
||
<li>$U(m\times m)$: 左奇异矩阵</li>
|
||
<li>$\Sigma(m\times n)$: 奇异值矩阵</li>
|
||
<li>$V(n\times n)$: 右奇异矩阵</li>
|
||
</ul>
|
||
<p>重点来了，SVD的意思就是，把一个本来由矩阵$A$表示的变换，转化成一个由$U,\Sigma,V$表示的变换。这个变换是，把一个向量，从以$V$为基向量的空间线性变换到成以$U$为基向量的空间中去（$\Sigma$的意义可以说是缩放，待证实，暂时忽略）。这样我们就可以更深入的理解这个变换了。</p>
|
||
<p>例如旋转$\vec{v_1}$90度得到$\vec{v_2}$，</p>
|
||
<p><img src="//davidz.cn/static/blog/2020-06-04-SVD-for-homogeneous-linear-equation/rotate90.svg" alt="rotate 90"></p>
|
||
<p>其中，</p>
|
||
<p>$$<br>
|
||
U=\begin{bmatrix}<br>
|
||
0 & 1 \\\<br>
|
||
1 & 0<br>
|
||
\end{bmatrix}<br>
|
||
$$</p>
|
||
<p>$$<br>
|
||
\Sigma=\begin{bmatrix}<br>
|
||
1 & 1<br>
|
||
\end{bmatrix}<br>
|
||
$$</p>
|
||
<p>$$<br>
|
||
V=\begin{bmatrix}<br>
|
||
-1 & 0 \\\<br>
|
||
0 & 1<br>
|
||
\end{bmatrix}<br>
|
||
$$</p>
|
||
<p>也就是说，矩阵$A$可以被理解为，我们把一个向量$\vec{v_1}$，从以$\vec{e_1}=[-1, 0]^T,\vec{e_2}=[0, 1]^T$为基向量的空间线性变换到了以$\vec{e_1}=[0, 1]^T,\vec{e_2}=[1, 0]^T$为基向量的空间中。这个变换表现为旋转了90度。</p>
|
||
<h2 id="所以如何理解"><a class="header-anchor" href="#所以如何理解">¶</a>所以如何理解</h2>
|
||
<p>说回求齐次线性方程组$Ax=0$的解来。</p>
|
||
<p>按照矩阵的意义，我们这里要求的是，已知一个线性变换$A$，给定一个$\vec{x}$，使得线性变换后的结果为$\vec{0}$。</p>
|
||
<p>此时非常重要的是，如果$x=\vec{0}$，那一定成立，但是我们想找一个非平凡的解。</p>
|
||
<p>我们暂时不关心这个解是否存在，也就是说如果不存在，我们就找个最接近的（最小二乘法思想），我们直接使用SVD分解矩阵$A$，得到对应的$U,\Sigma,V$。</p>
|
||
<p>按照SVD的作用，我们现在可以说，矩阵$A$这个线性变换，把一个$\vec{x}$，从以$V$为基向量的空间线性变换到了以$U$为基向量的空间中，而我们想找，在以$V$为基向量的空间中，哪个向量会在投影后趋近于或者等于$\vec{0}$。更重要的是，我们只在乎这个向量的方向，而不在乎他的大小，因为它等于$\vec{0}$是个平凡解，这就像最小二乘法中，我们规定$|\vec{x}|=1$。</p>
|
||
<p>这时，答案就开始变得清晰了，因为我们想找的$\vec{x}$，应该就是$V$这组基向量中特异值$\sigma$最小的那一个$\vec{e_{min}}$，也就是说$\vec{x}=\vec{e_{min}}$。此时有两种情况，</p>
|
||
<ol>
|
||
<li>$\sigma=0$， 那么$\vec{x}$投影后的就是$\vec{0}$。</li>
|
||
<li>$\sigma\neq0$，那么$\vec{x}$投影后是使$A\vec{x}$最小的解。因为如果$\vec{x}\neq\vec{e_{min}}$，也就是说它偏离了$\vec{e_{min}}$，那么它一定由$\vec{e_{min}}$和另一个基向量线性组合，而无论怎么组合，$\sigma_{combine}\geq\sigma_{min}$。</li>
|
||
</ol>
|
||
<p>因此，我们求解$A\vec{x}=0$的过程就是，</p>
|
||
<ol>
|
||
<li>$U,\Sigma,V^T=SVD(A)$</li>
|
||
<li>$\vec{x}=V[:, -1]$</li>
|
||
</ol>
|
||
|
||
<!--[if lt IE 9]><script>document.createElement('audio');</script><![endif]-->
|
||
<audio id="audio" loop="1" preload="auto" controls="controls"
|
||
data-autoplay="false">
|
||
<source type="audio/mpeg" src="https://davidz-blog.oss-cn-beijing.aliyuncs.com/music/Delacey - Dream It Possible.mp3">
|
||
</audio>
|
||
|
||
|
||
<div class="post-nav">
|
||
<hr>
|
||
|
||
<div class="post-nav-item">上一篇:<a href="/multiline-equation-in-hexo/" rel="prev"
|
||
title="技巧速记：如何在Hexo中插入多行公式">技巧速记：如何在Hexo中插入多行公式
|
||
</a></div>
|
||
|
||
|
||
<div class="post-nav-item">下一篇:<a href="/rattle-change-language/" rel="next"
|
||
title="技巧速记：更改 Rattle 的语言">技巧速记：更改 Rattle 的语言</a></div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
<div id='gitalk-container' class="comment link"
|
||
data-ae='true'
|
||
data-ci='489076c5dd3f5ba13f67'
|
||
data-cs='d6e3b245787b0b74d0dbe2639ef87f452a401194'
|
||
data-r='blog.davidz.cn'
|
||
data-o='DavidZhang73'
|
||
data-a='DavidZhang73'
|
||
data-d=''
|
||
data-p='https://cors-anywhere.azm.workers.dev/https://github.com/login/oauth/access_token'
|
||
>留言</div>
|
||
|
||
|
||
|
||
|
||
</div>
|
||
<div class="sidebar">
|
||
<div class="box animated fadeInRight">
|
||
<div class="subbox">
|
||
<img src="https://davidz-blog.oss-cn-beijing.aliyuncs.com/img/2019-1599483796.jpg" height=300 width=300></img>
|
||
<p>DavidZ</p>
|
||
<span>凡事都要留几分</span>
|
||
<dl>
|
||
|
||
|
||
</dl>
|
||
</div>
|
||
<ul>
|
||
<li><a href="/">15 <p>文章</p></a></li>
|
||
<li><a href="/categories">8 <p>分类</p></a></li>
|
||
<li><a href="/tags">15 <p>标签</p></a></li>
|
||
</ul>
|
||
</div>
|
||
|
||
|
||
|
||
<div class="box sticky animated fadeInRight faster">
|
||
<div id="toc" class="subbox">
|
||
<h4>目录</h4>
|
||
<ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#矩阵有何意义"><span class="toc-number">1.</span> <span class="toc-text">矩阵有何意义</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#SVD到底干了什么"><span class="toc-number">2.</span> <span class="toc-text">SVD到底干了什么</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#所以如何理解"><span class="toc-number">3.</span> <span class="toc-text">所以如何理解</span></a></li></ol>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
</div>
|
||
</div>
|
||
<div id="back-to-top" class="animated fadeIn faster">
|
||
<div class="flow"></div>
|
||
<span class="percentage animated fadeIn faster">0%</span>
|
||
<span class="iconfont icon-top02 animated fadeIn faster"></span>
|
||
</div><!-- hexo-inject:begin --><!-- hexo-inject:end -->
|
||
</body>
|
||
<footer>
|
||
<p class="copyright" id="copyright">
|
||
© 2022
|
||
<span class="gradient-text">
|
||
DavidZ
|
||
</span>.
|
||
Powered by <a href="http://hexo.io/" title="Hexo" target="_blank" rel="noopener">Hexo</a>
|
||
Theme
|
||
<span class="gradient-text">
|
||
<a href="https://github.com/TriDiamond/hexo-theme-obsidian" title="Obsidian" target="_blank" rel="noopener">Obsidian</a>
|
||
</span>
|
||
<small><a href="https://github.com/TriDiamond/hexo-theme-obsidian/blob/master/CHANGELOG.md" title="v1.4.7" target="_blank" rel="noopener">v1.4.7</a></small>
|
||
|
||
</br>
|
||
鲁ICP备
|
||
<span class="gradient-text">
|
||
<a href="https://beian.miit.gov.cn/" title="19008089号-1" target="_blank" rel="noopener">19008089号-1</a>
|
||
</span>
|
||
|
||
|
||
</p>
|
||
</footer>
|
||
|
||
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||
</script>
|
||
<script>
|
||
MathJax.Hub.Config({
|
||
"HTML-CSS": {
|
||
preferredFont: "TeX",
|
||
availableFonts: ["STIX", "TeX"],
|
||
linebreaks: {
|
||
automatic: true
|
||
},
|
||
EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50)
|
||
},
|
||
tex2jax: {
|
||
inlineMath: [
|
||
["$", "$"],
|
||
["\\(", "\\)"]
|
||
],
|
||
processEscapes: true,
|
||
ignoreClass: "tex2jax_ignore|dno",
|
||
skipTags: ['script', 'noscript', 'style', 'textarea', 'pre', 'code']
|
||
},
|
||
TeX: {
|
||
noUndefined: {
|
||
attributes: {
|
||
mathcolor: "red",
|
||
mathbackground: "#FFEEEE",
|
||
mathsize: "90%"
|
||
}
|
||
},
|
||
Macros: {
|
||
href: "{}"
|
||
}
|
||
},
|
||
messageStyle: "none"
|
||
});
|
||
</script>
|
||
<script>
|
||
function initialMathJax() {
|
||
MathJax.Hub.Queue(function () {
|
||
var all = MathJax.Hub.getAllJax(),
|
||
i;
|
||
// console.log(all);
|
||
for (i = 0; i < all.length; i += 1) {
|
||
all[i].SourceElement().parentNode.className += ' has-jax';
|
||
}
|
||
});
|
||
}
|
||
|
||
function reprocessMathJax() {
|
||
if (typeof MathJax !== 'undefined') {
|
||
MathJax.Hub.Queue(["Typeset", MathJax.Hub]);
|
||
}
|
||
}
|
||
</script>
|
||
|
||
|
||
|
||
<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/gitalk@1.6.2/dist/gitalk.min.css">
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/gitalk@1.6.2/dist/gitalk.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/jquery@3.4.1/dist/jquery.min.js"></script>
|
||
<script src="/js/plugin.js"></script>
|
||
<script src="/js/obsidian.js"></script>
|
||
<script src="/js/jquery.truncate.js"></script>
|
||
<script src="/js/search.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/typed.js@2.0.10/lib/typed.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/blueimp-md5@2.12.0/js/md5.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/social-share.js@1.0.16/dist/js/social-share.min.js"></script>
|
||
|
||
|
||
<script src="https://cdn.jsdelivr.net/npm/codemirror@5.45.0/lib/codemirror.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/javascript/javascript.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/css/css.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/xml/xml.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/htmlmixed/htmlmixed.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/clike/clike.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/php/php.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/shell/shell.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/python/python.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/cmake/cmake.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/powershell/powershell.min.js"></script>
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/codemirror@5.45.0/mode/yaml/yaml.min.js"></script>
|
||
|
||
<script src="/js/busuanzi.min.js"></script>
|
||
|
||
<script>
|
||
$(document).ready(function () {
|
||
if ($('span[id^="busuanzi_"]').length) {
|
||
initialBusuanzi();
|
||
}
|
||
});
|
||
</script>
|
||
|
||
<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/photoswipe.min.css">
|
||
<link rel="stylesheet" href="//cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/default-skin/default-skin.min.css">
|
||
|
||
|
||
<script src="//cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/photoswipe.min.js"></script>
|
||
<script src="//cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/photoswipe-ui-default.min.js"></script>
|
||
|
||
|
||
<!-- Root element of PhotoSwipe. Must have class pswp. -->
|
||
<div class="pswp" tabindex="-1" role="dialog" aria-hidden="true">
|
||
<!-- Background of PhotoSwipe.
|
||
It's a separate element as animating opacity is faster than rgba(). -->
|
||
<div class="pswp__bg"></div>
|
||
<!-- Slides wrapper with overflow:hidden. -->
|
||
<div class="pswp__scroll-wrap">
|
||
<!-- Container that holds slides.
|
||
PhotoSwipe keeps only 3 of them in the DOM to save memory.
|
||
Don't modify these 3 pswp__item elements, data is added later on. -->
|
||
<div class="pswp__container">
|
||
<div class="pswp__item"></div>
|
||
<div class="pswp__item"></div>
|
||
<div class="pswp__item"></div>
|
||
</div>
|
||
<!-- Default (PhotoSwipeUI_Default) interface on top of sliding area. Can be changed. -->
|
||
<div class="pswp__ui pswp__ui--hidden">
|
||
<div class="pswp__top-bar">
|
||
<!-- Controls are self-explanatory. Order can be changed. -->
|
||
<div class="pswp__counter"></div>
|
||
<button class="pswp__button pswp__button--close" title="Close (Esc)"></button>
|
||
<button class="pswp__button pswp__button--share" title="Share"></button>
|
||
<button class="pswp__button pswp__button--fs" title="Toggle fullscreen"></button>
|
||
<button class="pswp__button pswp__button--zoom" title="Zoom in/out"></button>
|
||
<!-- Preloader demo http://codepen.io/dimsemenov/pen/yyBWoR -->
|
||
<!-- element will get class pswp__preloader--active when preloader is running -->
|
||
<div class="pswp__preloader">
|
||
<div class="pswp__preloader__icn">
|
||
<div class="pswp__preloader__cut">
|
||
<div class="pswp__preloader__donut"></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="pswp__share-modal pswp__share-modal--hidden pswp__single-tap">
|
||
<div class="pswp__share-tooltip"></div>
|
||
</div>
|
||
<button class="pswp__button pswp__button--arrow--left" title="Previous (arrow left)">
|
||
</button>
|
||
<button class="pswp__button pswp__button--arrow--right" title="Next (arrow right)">
|
||
</button>
|
||
<div class="pswp__caption">
|
||
<div class="pswp__caption__center"></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async src="//www.googletagmanager.com/gtag/js?id=UA-157733505-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag() {
|
||
dataLayer.push(arguments);
|
||
}
|
||
gtag('js', new Date());
|
||
|
||
gtag('config', 'UA-157733505-1');
|
||
</script>
|
||
|
||
|
||
<script>
|
||
function initialTyped() {
|
||
var typedTextEl = $('.typed-text');
|
||
if (typedTextEl && typedTextEl.length > 0) {
|
||
var typed = new Typed('.typed-text', {
|
||
strings: ['凡事都要留几分', '颜值是第一生产力'],
|
||
typeSpeed: 90,
|
||
loop: true,
|
||
loopCount: Infinity,
|
||
backSpeed: 20,
|
||
});
|
||
}
|
||
}
|
||
|
||
if ($('.article-header') && $('.article-header').length) {
|
||
$(document).ready(function () {
|
||
initialTyped();
|
||
});
|
||
}
|
||
</script>
|
||
|
||
|
||
|
||
|
||
<!-- 例：百度统计 --> <script> var _hmt = _hmt || []; (function() { var hm = document.createElement("script"); hm.src = "https://hm.baidu.com/hm.js?your_code"; var s = document.getElementsByTagName("script")[0]; s.parentNode.insertBefore(hm, s); })(); </script>
|
||
|
||
</html>
|