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 想要了解矩阵快速幂，就不得不提到矩阵的概念。矩阵就像一个二维数组，存储了一组数据，如： \[M&#x3D;\left[
  \begin{matrix}
   1&amp;2&amp;3\\
   4&amp;5&amp;6\\
   7&amp;8&amp;9\end{matrix}
 \right]\] 用 \(M_{i,j}\) 表示矩阵第 \(i\) 行第 \(j\) 列的数据，如 \(M_"><meta property="og:type" content="article"><meta property="og:title" content="矩阵快速幂简述"><meta property="og:url" content="https://schtonn.github.io/blog/posts/matrix-pow/index.html"><meta property="og:site_name" content="schtonn"><meta property="og:description" content="引入
 想要了解矩阵快速幂，就不得不提到矩阵的概念。矩阵就像一个二维数组，存储了一组数据，如： \[M&#x3D;\left[
  \begin{matrix}
   1&amp;2&amp;3\\
   4&amp;5&amp;6\\
   7&amp;8&amp;9\end{matrix}
 \right]\] 用 \(M_{i,j}\) 表示矩阵第 \(i\) 行第 \(j\) 列的数据，如 \(M_"><meta property="og:locale" content="en_US"><meta property="article:published_time" content="2020-03-21T11:20:08.000Z"><meta property="article:modified_time" content="2022-10-19T15:02:06.666Z"><meta property="article:author" content="Alex"><meta property="article:tag" content="math"><meta property="article:tag" content="matrix"><meta name="twitter:card" content="summary"><link rel="canonical" href="https://schtonn.github.io/blog/posts/matrix-pow/"><script id="page-configurations">CONFIG.page={sidebar:"",isHome:!1,isPost:!0,lang:"en"}</script><title>矩阵快速幂简述 | schtonn</title><noscript><style>.sidebar-inner,.use-motion .brand,.use-motion .collection-header,.use-motion .comments,.use-motion .menu-item,.use-motion .pagination,.use-motion .post-block,.use-motion .post-body,.use-motion .post-header{opacity:initial}.use-motion .site-subtitle,.use-motion .site-title{opacity:initial;top:initial}.use-motion .logo-line-before i{left:initial}.use-motion .logo-line-after i{right:initial}</style></noscript></head><body itemscope itemtype="http://schema.org/WebPage"><div class="container use-motion"><div class="headband"></div><header class="header" itemscope itemtype="http://schema.org/WPHeader"><div class="header-inner"><div class="site-brand-container"><div class="site-nav-toggle"><div class="toggle" aria-label="Toggle navigation bar"><span class="toggle-line toggle-line-first"></span><span class="toggle-line toggle-line-middle"></span><span class="toggle-line toggle-line-last"></span></div></div><div class="site-meta"><a href="/blog/" class="brand" rel="start"><span class="logo-line-before"><i></i></span><h1 class="site-title">schtonn</h1><span class="logo-line-after"><i></i></span></a><p class="site-subtitle" itemprop="description">schtonn</p></div><div class="site-nav-right"><div class="toggle popup-trigger"></div></div></div><nav class="site-nav"><ul id="menu" class="menu"><li class="menu-item 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itemprop="name" content="Alex"><meta itemprop="description" content="blog"></span><script type="text/javascript" src="/blog/js/md5.js"></script><script></script><script>document.oncopy=function(e){window.event&&(e=window.event);try{var t=e.srcElement;return"INPUT"==t.tagName&&"text"==t.type.toLowerCase()||"TEXTAREA"==t.tagName}catch(e){return!1}}</script><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="schtonn"></span><header class="post-header"><h1 class="post-title" itemprop="name headline"> 矩阵快速幂简述</h1><div class="post-meta"><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-o"></i></span> <span class="post-meta-item-text">Posted on</span> <time title="Created: 2020-Mar-21 19:20:08" itemprop="dateCreated datePublished" datetime="2020-03-21T19:20:08+08:00">2020-Mar-21</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-check-o"></i></span> <span class="post-meta-item-text">Edited on</span> <time title="Modified: 2022-Oct-19 23:02:06" itemprop="dateModified" datetime="2022-10-19T23:02:06+08:00">2022-Oct-19</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-comment-o"></i></span> <span class="post-meta-item-text">Valine:</span><a title="valine" href="/blog/posts/matrix-pow/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/blog/posts/matrix-pow/" itemprop="commentCount"></span></a></span></div></header><div class="post-body" itemprop="articleBody"><h2 id="引入">引入</h2><p>想要了解矩阵快速幂，就不得不提到矩阵的概念。矩阵就像一个二维数组，存储了一组数据，如： <span class="math display">\[M=\left[ \begin{matrix} 1&amp;2&amp;3\\ 4&amp;5&amp;6\\ 7&amp;8&amp;9\end{matrix} \right]\]</span> 用 <span class="math inline">\(M_{i,j}\)</span> 表示矩阵第 <span class="math inline">\(i\)</span> 行第 <span class="math inline">\(j\)</span> 列的数据，如 <span class="math inline">\(M_{2,3}=6\)</span>。</p><a id="more"></a><h2 id="矩阵运算">矩阵运算</h2><h3 id="加法">加法</h3><p>矩阵的加法和实数加法类似，要求运算的两个矩阵大小相等，将对应位置相加即可。</p><h3 id="减法">减法</h3><p>与加法相同，对应位置相减即可。</p><h3 id="乘法">乘法</h3><p>矩阵乘法并非对应位置相乘。</p><p>若矩阵 <span class="math inline">\(A\times B=C\)</span> 有意义，那么 <span class="math inline">\(A\)</span> 的列数要求与 <span class="math inline">\(B\)</span> 的行数相等。</p><p>若 <span class="math inline">\(A\)</span> 的列数与 <span class="math inline">\(B\)</span> 的行数等于 <span class="math inline">\(n\)</span>：</p><p><span class="math display">\[C_{i,j}=\sum_{k=1}^{n}A_{i,k}\times B_{k,j}\]</span></p><p>即结果第 <span class="math inline">\(i\)</span> 行第 <span class="math inline">\(j\)</span> 列的值为 <span class="math inline">\(A\)</span> 的第 <span class="math inline">\(i\)</span> 行与 <span class="math inline">\(B\)</span> 的第 <span class="math inline">\(j\)</span> 列的<strong>对应项相乘后求和</strong>。</p><p>一个很直观的理解是，把矩阵 <span class="math inline">\(A\)</span> 放在结果的左侧， 矩阵 <span class="math inline">\(B\)</span> 放在结果的上方，将行与列延长，对应相乘后结果（此例中为 <span class="math inline">\((7\times2)+(8\times5)+(9\times8)\)</span>）写在交点处。 <span class="math display">\[ \begin{array}{clr} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \left[ \begin{matrix} 1&amp;\bold{2}&amp;3\\ 4&amp;\bold{5}&amp;6\\ 7&amp;\bold{8}&amp;9\end{matrix} \right]\\ \left[ \begin{matrix} 1&amp;2&amp;3\\ 4&amp;5&amp;6\\ \bold{7}&amp;\bold{8}&amp;\bold{9}\end{matrix} \right]\left[ \begin{matrix} &amp;\downarrow&amp;&amp;\\ &amp;\downarrow&amp;&amp;\\ \rightarrow&amp;126&amp;&amp;\end{matrix} \right] \end{array}\]</span></p><p>注意矩阵乘法满足结合律，但是<strong>不一定满足交换律</strong>，<span class="math inline">\(A\times B\)</span> 成立不一定代表 <span class="math inline">\(B\times A\)</span> 成立。</p><h3 id="除法">除法</h3><p><span class="math inline">\(\cdots\)</span> 我不会，长大后再学习（反正就是乘法的逆运算）。</p><h2 id="快速幂">快速幂</h2><p>快速幂详见<a href="/blog/posts/quick-pow">另一篇文章</a>。</p><h2 id="矩阵快速幂">矩阵快速幂</h2><p>由于矩阵的幂符合快速幂所需的性质，所以可以使用直观地直接进行矩阵快速幂，方法和普通快速幂一样，但是要注意乘法顺序。<code>ans</code>的初始值要特别注意。如果想让它充当1，则需要设置成相同阶数的单位矩阵 <span class="math inline">\(I\)</span>。 <span class="math display">\[ I=\left[\begin{matrix}1&amp;0&amp;0&amp;\cdots&amp;0\\0&amp;1&amp;0&amp;\cdots&amp;0\\0&amp;0&amp;1&amp;\cdots&amp;0\\\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;0\\0&amp;0&amp;0&amp;0&amp;1\end{matrix}\right] \]</span></p><h3 id="递推">递推</h3><p>使用矩阵快速幂，可以用 <span class="math inline">\(O(\log n)\)</span> 的时间复杂度完成 <span class="math inline">\(n\)</span> 次线性齐次递推，如斐波那契数列。矩阵可以通过递推关系推算出来。一般来说，这里的递推关系用一个列向量来记录运算的过程值，然后通过左乘一个系数矩阵，<strong>转移</strong>到下一个状态，在这里就是转移到数列的下一位。也就是说：</p><p><span class="math display">\[\left[\begin{matrix}1&amp;1\\0&amp;1\end{matrix}\right] \times \left[\begin{matrix}F_{n-1}\\F_{n-2}\end{matrix}\right] = \left[\begin{matrix}F_n\\F_{n-1}\end{matrix}\right]\]</span> <span class="math display">\[\because F_1=F_2=1\]</span> <span class="math display">\[\therefore F_n=\left[\begin{matrix}1&amp;1\\0&amp;1\end{matrix}\right]^n \times \left[\begin{matrix}1\\1\end{matrix}\right]\]</span></p><p>在这里，一次乘法对应着状态前进一步。</p><h3 id="推算矩阵">推算矩阵</h3><p>系数矩阵由转移关系得来。以斐波那契数列为例：</p><p><span class="math display">\[\because F_n=1F_{n-1}+1F_{n-2}\]</span> <span class="math display">\[F_{n-1}=1F_{n-1}+0F_{n-2}\]</span> <span class="math display">\[\therefore M=\left[\begin{matrix}1&amp;1\\1&amp;0\end{matrix}\right]\]</span></p><p>如果直接带入矩阵乘法，你会发现这样转移完全合理，甚至惊人地简洁。</p><h3 id="代码斐波那契数列">代码（斐波那契数列）</h3><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">"bits/stdc++.h"</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N=<span class="number">90</span>;</span><br><span class="line"><span class="keyword">int</span> n;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">matrix</span>&#123;</span></span><br><span class="line">    <span class="keyword">int</span> v[N+<span class="number">2</span>][N+<span class="number">2</span>];</span><br><span class="line">    <span class="keyword">int</span> x,y;</span><br><span class="line">&#125;t,I,ans;</span><br><span class="line"><span class="keyword">int</span> pre[N+<span class="number">2</span>];</span><br><span class="line">ostream&amp;<span class="keyword">operator</span>&lt;&lt;(ostream&amp;ous,matrix a)&#123;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;a.x;i++)&#123;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">0</span>;j&lt;a.y;j++)&#123;</span><br><span class="line">            ous&lt;&lt;a.v[i][j]&lt;&lt;<span class="string">' '</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        ous&lt;&lt;<span class="built_in">endl</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">return</span> ous;</span><br><span class="line">&#125;</span><br><span class="line">matrix <span class="keyword">operator</span>+(matrix a,matrix b)&#123;</span><br><span class="line">    matrix c;</span><br><span class="line">    <span class="keyword">if</span>(a.x!=b.x||a.y!=b.y)<span class="keyword">return</span> c;</span><br><span class="line">    <span class="keyword">int</span> x=a.x,y=a.y;</span><br><span class="line">    c.x=x;</span><br><span class="line">    c.y=y;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;x;i++)&#123;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">0</span>;j&lt;y;j++)&#123;</span><br><span class="line">            c.v[i][j]=(a.v[i][j]+b.v[i][j]);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">return</span> c;</span><br><span class="line">&#125;</span><br><span class="line">matrix <span class="keyword">operator</span>*(matrix a,matrix b)&#123;</span><br><span class="line">    matrix c;</span><br><span class="line">    <span class="keyword">if</span>(a.y!=b.x)<span class="keyword">return</span> c;</span><br><span class="line">    <span class="keyword">int</span> x=a.x,y=b.y,z=a.y;</span><br><span class="line">    c.x=x;</span><br><span class="line">    c.y=y;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;x;i++)&#123;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">0</span>;j&lt;y;j++)&#123;</span><br><span class="line">            c.v[i][j]=<span class="number">0</span>;</span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> k=<span class="number">0</span>;k&lt;z;k++)&#123;</span><br><span class="line">                c.v[i][j]=(c.v[i][j]+a.v[i][k]*b.v[k][j]);</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">return</span> c;</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">init</span><span class="params">()</span></span>&#123;</span><br><span class="line">    t.x=t.y=<span class="number">2</span>;</span><br><span class="line">    I.x=I.y=<span class="number">2</span>;</span><br><span class="line">    t.v[<span class="number">0</span>][<span class="number">0</span>]=t.v[<span class="number">0</span>][<span class="number">1</span>]=t.v[<span class="number">1</span>][<span class="number">0</span>]=<span class="number">1</span>;</span><br><span class="line">    I.v[<span class="number">0</span>][<span class="number">0</span>]=I.v[<span class="number">1</span>][<span class="number">1</span>]=<span class="number">1</span>;</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>&#123;</span><br><span class="line">    <span class="built_in">cin</span>&gt;&gt;n;</span><br><span class="line">    init();</span><br><span class="line">    <span class="keyword">int</span> cnt=n;</span><br><span class="line">    ans=I;</span><br><span class="line">    <span class="keyword">while</span>(cnt)&#123;</span><br><span class="line">        <span class="keyword">if</span>(cnt&amp;<span class="number">1</span>)ans=ans*t;</span><br><span class="line">        t=t*t;</span><br><span class="line">        cnt&gt;&gt;=<span class="number">1</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="built_in">cout</span>&lt;&lt;ans.v[<span class="number">0</span>][<span class="number">0</span>]&lt;&lt;<span class="built_in">endl</span>;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></div><div><ul class="post-copyright"><li class="post-copyright-author"> <strong>Post author:</strong> Alex</li><li class="post-copyright-link"> <strong>Post link:</strong> <a href="https://schtonn.github.io/blog/posts/matrix-pow/" title="矩阵快速幂简述">https://schtonn.github.io/blog/posts/matrix-pow/</a></li><li class="post-copyright-license"> <strong>Copyright Notice:</strong> All articles in this blog are licensed under<a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" rel="noopener" target="_blank"><i class="fa fa-fw fa-creative-commons"></i> BY-NC-SA</a> unless stating additionally.</li></ul></div><footer class="post-footer"><div class="post-tags"><a href="/blog/tags/math/" rel="tag"><i class="fa fa-tag"></i> math</a><a href="/blog/tags/matrix/" rel="tag"><i class="fa fa-tag"></i> matrix</a></div><div class="post-nav"><div class="post-nav-item"><a href="/blog/posts/segment-tree/" rel="prev" title="线段树"><i class="fa fa-chevron-left"></i> 线段树</a></div><div class="post-nav-item"> <a href="/blog/posts/hamilton/" rel="next" title="哈密尔顿回路">哈密尔顿回路<i class="fa fa-chevron-right"></i></a></div></div></footer></article></div><div class="comments" 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