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cxy0714 · cxy0714 · commit 18e4b1bd229f · 2025-01-02T20:36:18.000+08:00
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 ---
+title: "About"
 date: "2022-08-29T21:48:51-07:00"
-title: About
 ---
-# 2024.11.03
-新家正在施工中
+# 2025.01.02
+I'm Xingyu Chen(陈星宇), a second-year Ph.D. candidate in Statistics at [School of Mathematical Sciences](https://math.sjtu.edu.cn/Default/index/), [Shanghai Jiao Tong University(SJTU)](https://www.sjtu.edu.cn/), now interested in causal inference, higher order influence function, bootstrap and statistical computing. I am very fortunate to be advised by [Professor Lin Liu (刘林)](https://linliu-stats.github.io/). 
+
+Before joining SJTU, I received my B.S. degree in mathematics and applied mathematics from [School of Mathematical Sciences](https://maths.hust.edu.cn/), [Huazhong University of Science and Technology(HUST)](http://www.hust.edu.cn/). 
+
+⚡ Fun fact: "0714" is the discipline code for Statistics in Chinese Ministry of Education's subject classification system, where "07" represents Science and "0701" represents Mathematics. ([Source](https://www.cdgdc.edu.cn/dslxkpgjggb/))
 # 2022.08.29
 
 {{<figure src="/media/images/about_my_photo.jpg" caption=" " width="640">}} 
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 ---
-title: Mathematica踩坑-矩阵符号计算 | 陈星宇
+title: Mathematica踩坑-矩阵符号计算
 date: 2023-03-17T18:34:35+08:00
 updated: 2023-04-19T19:11:00+08:00 
 author: 陈星宇
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+    
+
+    <h1 class="article-title">Mathematica踩坑-矩阵符号计算</h1>
+
+    
+    <span class="article-date">2023-03-17</span>
+    
+
+    <div class="article-content">
+      
+      <h2 id="问题由来">问题由来</h2>
+<p>最近为了推导公式，需要验证一些（20多个）高维矩阵的内积结果（手算1-8h不等），而且需要一个靠谱的帮助我检验计算结果的工具。所以想到了Mathematica。</p>
+<p>我的计算大概是：$k^2\times k^5$的两个矩阵$A,B$做内积：$Tr(A \cdot B^T)$。通过$Tr$的可交换性质可以转化为$1 \times k^2$的行向量与$k^2\times k^5$的矩阵与$k^5\times 1$的列向量一个&quot;内积&quot;，所以就是一堆（$k^2\times k^5$）数相加，不过矩阵$A,B$都不是标量矩阵，元素都是符号，维数也还会随公式的阶数增高。</p>
+<p>下面是矩阵大概的形式（还有更长的），其中$Dimensions[\eta_2^{-1}]=k\times k,Dimensions[\eta_1]=k\times 1$。$I_k$是$k$阶单位阵，$K_{k,k}$是$k^2\times k^2$矩阵，学名叫作交换矩阵（Commutation Matrix），在矩阵向量化操作和矩阵张量积里是一个重要的工具，可以简单看<a href="https://blog.csdn.net/weixin_39274659/article/details/113747158">交换矩阵 commutation matrix：理论与matlab仿真_B417科研笔记的博客-CSDN博客</a>来了解，要了解更多Commutation Matrix可以搜英文。</p>
+<p>$$vec^{T}(\eta_{2}^{-1}\eta_{1}\eta_{1}^{T}\eta_{2}^{-1})\otimes\eta_{2}^{-1}\otimes\eta_{2}^{-1}\cdot I_{k}\otimes K_{k,k}\otimes I_{k}$$</p>
+<p>还有这个$vec^T \eta_2^{-1}$中的$vec():\mathbb{R}^{p\times q} \rightarrow \mathbb{R}^{pq\times1}$算子，就是把矩阵重新排序成为列向量，&ldquo;拉直&rdquo;，这个是定义矩阵导数的工具。</p>
+<p>（关于矩阵对矩阵求导数：考虑矩阵函数对矩阵变元求导，得到的导数的元素如何排列？一个想法就是把矩阵都拉成向量，这样就化为了向量函数对向量变元求导，这是微积分都很熟悉的，有一些好的排布规则，并且可以证明，这种形式的矩阵导数有比较好的性质，比如简便的链式法则，以及$X$对自己求导时得到的是一个单位阵。另一个想法就是直接像矩阵张量积一样的排布导数，首先你可以想象一下$X$对自己求导的结果，它甚至不是一个单位阵，不过这只是槽点之一，我还没有了解更多。更多信息可以看这本书的$\S1.4$节：<a href="https://link.springer.com/book/10.1007/1-4020-3419-9">Advanced Multivariate Statistics with Matrices | SpringerLink</a>，本书也介绍了vec和Commutation Matrix的性质，比较全面。）</p>
+<p>具体地，$vec(A)=vec([a_1,a_2,…,a_n])=[a_1^T,a_2^T,…,a_n^T]^T$。</p>
+<p>矩阵张量积（Tensor Product）是$A\otimes B=[a_{i,j}B]_{(i,j)}$。也叫克罗内克积（Kronecker product），外积，直积。</p>
+<p>总之我的矩阵里充斥着矩阵张量积，vec操作和交换矩阵$K_{p,q}$。同时我预测出了这些矩阵内积的结果，我希望比较一下他们是否相等，检验我的预测。</p>
+<h3 id="mathematica">Mathematica</h3>
+<p>我在大一寒假时看过一些Mathematica的文章和视频，之后用它做了一些大学物理实验的一些公式的计算？但是似乎纯粹当计算器用的，没有太多复杂的操作，等于熟悉了一下基本的文档操作吧。（不小心全给删除清空了，所以记不得到底干过啥了。）但是我也看过许多他的例子，精美的例子之类的，对他友好的帮助文档印象很深，还有就是清华大学数学科学学院<a href="http://blog.siqiliu.com/cn/index.html">刘思齐老师</a>的<a href="https://www.bilibili.com/video/BV1av411N7Xi/?spm_id_from=333.999.0.0&amp;vd_source=d604f008cde1c2b512c49f045d95e4cd">Mathematica网课</a>很好。</p>
+<p>之前我下载了Mathematica 13.0的中文破解版，然后现在要用它的时候发现我需要的内置函数在13.1版本才出现了，这时候再去搜索一圈，已经找不到免费的资源了，全是加微信公众号关注，xx币购买链接云云，一直就很反感这些东西。</p>
+<p>所以我找到了官网，发现新用户可以免费试用15天，但是直接下载电脑版还需要一些操作，就没有下载pc版，发现Mathematica有云系统，可以在线编译，存储云文件，网站:<a href="https://mathematica.wolframcloud.com/">Wolfram Mathematica (wolframcloud.com)</a>。在线编译已经完全可以满足我的需要了。而且我的运算量也不太大，跑代码基本都是秒出。但是可以帮我省去几十个小时的计算了！</p>
+<h2 id="写代码时碰到的问题">写代码时碰到的问题</h2>
+<h3 id="mathematica中的矩阵和行向量列向量">Mathematica中的矩阵和行向量、列向量</h3>
+<p>Mathematica帮助文档里有<a href="https://reference.wolfram.com/language/guide/MatricesAndLinearAlgebra.html">矩阵和线性代数—Wolfram 语言参考资料</a>。</p>
+<p>Mathematica中定义向量和矩阵的内置函数有Table[],Array[],也可以用来构造向量。但是我一开始却始终搞不清楚他们的维数怎么回事。现在我明白了，<strong>两个花括号<code>{{}}</code>包住的就是矩阵</strong>，<strong>一个花括号<code>{}</code>就是Mathematica里的向量</strong>，Mathematica里的向量<strong>既不是行向量，也不是列向量</strong>，如果向量在运算中放在右边就是列向量，放在左边就是行向量，你使用转置函数Transpose[]也没用。所以这个问题让我一开始搞得一头雾水。算矩阵乘法时不知道哪里出了错。</p>
+<p>如果你需要定义列向量，请这样定义：</p>
+<p><code>T=Array[b,{2,1}]</code></p>
+<p>这样他就是两个花括号包裹了，将其视为$2\times 1$的矩阵，即列向量。</p>
+<p>另一个问题是，在<strong>定义矩阵时不要在后面加上//MartixForm</strong>，一般在后面加上这个语句可以让你的输出变成好看的矩阵形式（否则会输出<code>{{}}</code>的列表形式，看起来不方便），但是如果你在定义里加上了这个，后面的矩阵计算会出现各种问题，这是我一开始矩阵运算失败的第二个问题，然后绝望之下在听清华数院的刘思齐老师的一个Mathematica网课中，他提到了这一句：<a href="https://www.bilibili.com/video/BV1av411N7Xi?p=9&amp;vd_source=d604f008cde1c2b512c49f045d95e4cd">lecture-2-4_哔哩哔哩_bilibili</a>，在我心里救了Mathematica一命。</p>
+<p>另外用Dimensions[]函数可以查看矩阵（多少个花括号都行）的维数。</p>
+<p>比如$p\times q$的矩阵被Dimensions[]作用之后结果是 <code>{p,q}</code>。</p>
+<h3 id="矩阵张量积">矩阵张量积</h3>
+<p>矩阵张量积(外积，直积)这个在Mathematica里直接有函数KroneckerProduct[]。</p>
+<p>但是一开始我竟然没有搜索到KroneckerProduct[]，先是搜索到了Outer 外积，这个东西是张量外积，两个向量（1个花括号的）可以用这个得到矩阵，但是矩阵和矩阵用这个做外积，虽然形式上是一样的，但是它会出现4个花括号，用Dimensions[]看其维数，会得到类似这样的结果 <code>{2,2,2,2}</code>这已经不是矩阵的维数了。这个是我在计算中遇到的第三个问题，好在最后发现了他有正经的矩阵外积函数KroneckerProduct[]</p>
+<p>所以矩阵张量积积直接用函数KroneckerProduct[]。</p>
+<h3 id="vec算子的定义和交换矩阵k_pq的定义">vec算子的定义和交换矩阵$K_{p,q}$的定义</h3>
+<p>这两个函数的定义，我在一通搜索之后，置换矩阵PermutationMatrix[]的<strong>应用举例</strong>中：<a href="https://reference.wolfram.com/language/ref/PermutationMatrix.html">PermutationMatrix—Wolfram 语言参考资料</a>发现了。</p>
+<p><code>vec[m_?MatrixQ] := Flatten[m, {{2, 1}}]</code></p>
+<p>它将矩阵的列堆叠成单个向量。问题是这个向量是Mathematica中的向量，只有一个花括号，既不是行向量也不是列向量，所以这是我后面计算中遇到的第四个问题，但是现在我已经是个成熟的向量使用者了。我搜索了&quot;Mathematica怎么把向量变成矩阵&quot;之类的之后得到了回答。</p>
+<p>只需要在定义外边加上一个花括号即可<code>{}</code>。</p>
+<p><code>vec[m_?MatrixQ] := {Flatten[m, {{2, 1}}]}</code></p>
+<p>交换矩阵的定义可以直接拿他他的代码用，没有奇奇怪怪问题。但是你可以改个简短点的函数名。</p>
+<p><code>vecPermutationMatrix[p_, q_] :=  PermutationMatrix[Flatten[Partition[Range[p q], p], {{2, 1}}]]</code></p>
+<h3 id="比较两个矩阵内积是否相等">比较两个矩阵内积是否相等</h3>
+<p>这个是第五个问题，我本来以为直接相减，等于0就是相等就ok了。结果减出来都不是0，可是我可以确保我的计算是正确的。然后我使用了<strong>ExpandAll[]</strong>，讲所有括号去掉，展开，终于得到了0。</p>
+<p>问题可能是这样的，我的矩阵内积，有9个符号、12个符号相乘，然后再相加。它里面有大量合并同类型搞出来的括号之类的东西。比如：</p>
+<p><code>A=a (b+c+d) B=(a b+a c+a d)</code></p>
+<p><code>A-B=a (b+c+d)-(a b +a c+a d)</code></p>
+<p>就是虽然A和B一样，但是他没有展开，有很多括号，减出来他们不会抵消，还是就这样写出来，合并同类项。</p>
+<p>所以需要用ExpandAll[]，展开之后，没有括号了，他就可以合并同类型抵消得到0了。</p>
+<p><code>ExpandAll[A-B]=0</code></p>
+<h2 id="小技巧">小技巧</h2>
+<h3 id="quit-kernel">Quit Kernel</h3>
+<p>还有一个问题也是在刘思齐老师的Mathematica课里听到的，就是如果新建一个文件时，最好要Quit Kernel一下，这个按钮可以把之前的数据全给清楚，避免不必要的错误，有可能你的某个变量赋过了值，结果你把他忘记了。Quit Kernel在Evaluation列表之下。</p>
+<p>一个技巧就是未被赋值的变量是蓝色的，已被赋值的变量颜色是黑色的。</p>
+<h3 id="一键清除输出和导出pdf格式">一键清除输出和导出PDF格式</h3>
+<p>在程序编写完成后，为了输出一个好的PDF报告，可以先一键清除输出（Evaluation-&gt;Delete All Outputs），再Quit Kernel。然后计算所有单元（Evaluate All cells），再导出为PDF（File-&gt;Print to PDF）。</p>
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+            <a href="https://gohugo.io/" class="footer-links-kudos">Made with <img src="/images/hugo-logo.png" alt="Img link to Hugo website" width="22" height="22"></a>
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+    
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+<script src="/js/math-code.js"></script>
+<script async src="//cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML"></script>
+
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+  </body>
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+
diff --git a/public/2024/11/12/无数反复的涌动/index.html b/public/2024/11/12/无数反复的涌动/index.html
@@ -1,6 +1,6 @@
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 <html lang="en-US">
-  <head><script src="/livereload.js?mindelay=10&amp;v=2&amp;port=6958&amp;path=livereload" data-no-instant defer></script>
+  <head>
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 <meta name="viewport" content="width=device-width, initial-scale=1.0">
 <meta name="generator" content="Hugo 0.136.5">
@@ -10,7 +10,7 @@
 <meta property="og:title" content="无数反复的涌动 - A Hugo website">
 
 
-  <link href='//localhost:6958/favicon.ico' rel='icon' type='image/x-icon'/>
+  <link href='/favicon.ico' rel='icon' type='image/x-icon'/>
 
 
 
diff --git a/public/about/index.html b/public/about/index.html
@@ -1,6 +1,6 @@
 <!DOCTYPE html>
 <html lang="en-US">
-  <head><script src="/livereload.js?mindelay=10&amp;v=2&amp;port=6958&amp;path=livereload" data-no-instant defer></script>
+  <head>
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 <meta name="viewport" content="width=device-width, initial-scale=1.0">
 <meta name="generator" content="Hugo 0.136.5">
@@ -10,7 +10,7 @@
 <meta property="og:title" content="About - A Hugo website">
 
 
-  <link href='//localhost:6958/favicon.ico' rel='icon' type='image/x-icon'/>
+  <link href='/favicon.ico' rel='icon' type='image/x-icon'/>
 
 
 
@@ -64,8 +64,10 @@ <h1 class="article-title">About</h1>
 
     <div class="article-content">
       
-      <h1 id="20241103">2024.11.03</h1>
-<p>新家正在施工中</p>
+      <h1 id="20250102">2025.01.02</h1>
+<p>I&rsquo;m Xingyu Chen(陈星宇), a second-year Ph.D. candidate in Statistics at <a href="https://math.sjtu.edu.cn/Default/index/">School of Mathematical Sciences</a>, <a href="https://www.sjtu.edu.cn/">Shanghai Jiao Tong University(SJTU)</a>, now interested in causal inference, higher order influence function, bootstrap and statistical computing. I am very fortunate to be advised by <a href="https://linliu-stats.github.io/">Professor Lin Liu (刘林)</a>.</p>
+<p>Before joining SJTU, I received my B.S. degree in mathematics and applied mathematics from <a href="https://maths.hust.edu.cn/">School of Mathematical Sciences</a>, <a href="http://www.hust.edu.cn/">Huazhong University of Science and Technology(HUST)</a>.</p>
+<p>⚡ Fun fact: &ldquo;0714&rdquo; is the discipline code for Statistics in Chinese Ministry of Education&rsquo;s subject classification system, where &ldquo;07&rdquo; represents Science and &ldquo;0701&rdquo; represents Mathematics. (<a href="https://www.cdgdc.edu.cn/dslxkpgjggb/">Source</a>)</p>
 <h1 id="20220829">2022.08.29</h1>
 <figure><img src="/media/images/about_my_photo.jpg"
     alt="" width="640"><figcaption>
diff --git a/public/categories/index.html b/public/categories/index.html
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 <html lang="en-US">
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-  <link href='//localhost:6958/favicon.ico' rel='icon' type='image/x-icon'/>
+  <link href='/favicon.ico' rel='icon' type='image/x-icon'/>
 
 
 
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diff --git a/public/categories/个人经历/index.html b/public/categories/个人经历/index.html
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diff --git a/public/categories/程序/index.html b/public/categories/程序/index.html
diff --git a/public/index.html b/public/index.html
diff --git a/public/index.xml b/public/index.xml
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