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<article class="md-content__inner md-typeset">
<h1 id="_1">矩阵类型<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>特殊的矩阵结构能够解锁计算捷径和数学保证。本文涵盖单位矩阵、对角矩阵、对称矩阵、三角矩阵、正交矩阵、正定矩阵、稀疏矩阵和随机矩阵，这些类型出现在协方差估计、图算法、正则化和马尔可夫链中。</em></p>
<ul>
<li>
<p>并非所有矩阵都一样。不同的结构赋予矩阵特殊的性质，使它们计算更快、更易于推理，或两者兼得。以下是你最常遇到的类型。</p>
</li>
<li>
<p><strong>方阵</strong>的行数和列数相同（<span class="arithmatex">\(n \times n\)</span>）。大多数有趣的性质（行列式、特征值、逆）只适用于方阵。</p>
</li>
<li>
<p><strong>单位矩阵</strong> <span class="arithmatex">\(I\)</span> 是一个对角线为 1、其余为 0 的方阵。它是"什么都不做"的变换：<span class="arithmatex">\(AI = IA = A\)</span> 对任何兼容的矩阵 <span class="arithmatex">\(A\)</span>。</p>
</li>
</ul>
<div class="arithmatex">\[
I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
\]</div>
<ul>
<li>
<p><strong>零矩阵</strong> <span class="arithmatex">\(O\)</span> 的所有元素都为零。它将每个向量映射到零向量，破坏所有信息。</p>
</li>
<li>
<p><strong>对角矩阵</strong>除主对角线外全为零。将向量乘以对角矩阵只是独立地缩放每个分量，非常高效。</p>
</li>
</ul>
<div class="arithmatex">\[
D = \begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix}
\]</div>
<ul>
<li><strong>对称矩阵</strong>等于其转置：<span class="arithmatex">\(A = A^T\)</span>，意味着 <span class="arithmatex">\(A_{ij} = A_{ji}\)</span>。对称矩阵有一个特殊性质：它们的特征向量总是相互垂直。协方差矩阵总是对称的。</li>
</ul>
<div class="arithmatex">\[
S = \begin{bmatrix} 3 & -1 \\ -1 & 6 \end{bmatrix}
\]</div>
<ul>
<li><strong>三角矩阵</strong>在对角线的一侧全为零。<strong>下三角</strong>在上方全为零，<strong>上三角</strong>在下方全为零。它们对于通过前向或回代高效求解方程组至关重要。</li>
</ul>
<div class="arithmatex">\[
L = \begin{bmatrix} 2 & 0 & 0 \\ 1 & 3 & 0 \\ -1 & 2 & 4 \end{bmatrix} \qquad U = \begin{bmatrix} 5 & -1 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & -2 \end{bmatrix}
\]</div>
<ul>
<li>
<p>三角矩阵的行列式就是其对角线元素的乘积。</p>
</li>
<li>
<p><strong>正交矩阵</strong>具有转置等于逆的性质：<span class="arithmatex">\(Q^TQ = QQ^T = I\)</span>。</p>
</li>
<li>
<p>这意味着你只需转置就能"撤销"变换，计算成本很低。其列是标准正交的（单位长度且相互垂直）。</p>
</li>
<li>
<p><strong>稀疏矩阵</strong>的大多数元素为零，而<strong>稠密矩阵</strong>的大多数元素非零。</p>
</li>
</ul>
<p><img alt="稀疏 vs 稠密：点表示非零元素" src="../../images/sparse_dense.svg" /></p>
<ul>
<li>
<p>在实践中，许多现实世界的矩阵是极其稀疏的。</p>
</li>
<li>
<p>一个拥有百万用户的社交网络可以表示为一个 <span class="arithmatex">\(10^6 \times 10^6\)</span> 的矩阵，但每个人只连接到少数其他人，所以几乎所有元素都是零。</p>
</li>
</ul>
<p><img alt="一个小型社交网络及其邻接矩阵：大多数元素为零" src="../../images/social_network_matrix.svg" /></p>
<ul>
<li>
<p><strong>置换矩阵</strong>是通过重排单位矩阵的行得到的。乘以它会打乱向量的元素。每行每列恰好有一个 1，其余为 0。</p>
</li>
<li>
<p>例如，下面的矩阵将元素 3 移到位置 1，元素 1 移到位置 2，元素 2 移到位置 3：</p>
</li>
</ul>
<div class="arithmatex">\[
P = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}
\]</div>
<ul>
<li><strong>托普利茨矩阵</strong>沿每条对角线（左上到右下）具有相同的值。注意每条对角线是如何恒定的：</li>
</ul>
<div class="arithmatex">\[
T = \begin{bmatrix} a & b & c \\ d & a & b \\ e & d & a \end{bmatrix}
\]</div>
<ul>
<li>
<p>这种结构出现在信号处理和卷积中，因为将固定滤波器滑过信号等价于乘以托普利茨矩阵。</p>
</li>
<li>
<p><strong>循环矩阵</strong>是一种特殊的托普利茨矩阵，其中每一行是上一行的循环移位。当一行到达末尾时，它会绕回：</p>
</li>
</ul>
<div class="arithmatex">\[
C = \begin{bmatrix} 1 & 3 & 2 \\ 2 & 1 & 3 \\ 3 & 2 & 1 \end{bmatrix}
\]</div>
<ul>
<li>
<p>循环矩阵与离散傅里叶变换（DFT）密切相关，并且是循环卷积如何工作的核心。</p>
</li>
<li>
<p><strong>埃尔米特矩阵</strong>是对称矩阵在复数域中的等价形式：<span class="arithmatex">\(A = A^\ast\)</span>（其中 <span class="arithmatex">\(A^\ast\)</span> 是共轭转置）。</p>
</li>
<li>
<p>对于实值矩阵，埃尔米特矩阵和对称矩阵是一回事。你会在量子计算和信号处理中遇到它们。</p>
</li>
<li>
<p><strong>酉矩阵</strong>是正交矩阵在复数域中的等价形式：<span class="arithmatex">\(U^\ast U = UU^\ast = I\)</span>。正如正交矩阵在实空间中保持长度，酉矩阵在复空间中保持长度。</p>
</li>
<li>
<p><strong>幂等矩阵</strong>满足 <span class="arithmatex">\(A^2 = A\)</span>。应用变换两次等同于应用一次，这使得它成为一个<strong>投影</strong>。一旦你投影了，再次投影不会改变任何东西。</p>
</li>
<li>
<p><strong>幂零矩阵</strong>满足对某个幂次 <span class="arithmatex">\(k\)</span> 有 <span class="arithmatex">\(A^k = O\)</span>（零矩阵）。应用变换足够多次后，所有东西都坍缩为零。例如：</p>
</li>
</ul>
<div class="arithmatex">\[
\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}^2 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
\]</div>
<ul>
<li><strong>布尔矩阵</strong>（或二元矩阵）只包含 0 和 1。它表示是/否关系。例如，在一个有 3 个节点的图中，<strong>邻接矩阵</strong>记录哪些节点相连：</li>
</ul>
<div class="arithmatex">\[
B = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}
\]</div>
<ul>
<li>
<p>这里，节点 1 连接到节点 2 和 3，但节点 2 和 3 之间没有连接。</p>
</li>
<li>
<p><strong>范德蒙矩阵</strong>由一组值的连续幂次构成。给定值 <span class="arithmatex">\(x_1, x_2, x_3\)</span>：</p>
</li>
</ul>
<div class="arithmatex">\[
V = \begin{bmatrix} 1 & x_1 & x_1^2 \\ 1 & x_2 & x_2^2 \\ 1 & x_3 & x_3^2 \end{bmatrix}
\]</div>
<ul>
<li>
<p>这种结构出现在多项式插值中：找到通过给定点集的唯一多项式。</p>
</li>
<li>
<p><strong>海森堡矩阵</strong>是"几乎"三角的，在第一次次对角线以下全为零：</p>
</li>
</ul>
<div class="arithmatex">\[
H = \begin{bmatrix} 4 & 2 & 1 \\ 3 & 5 & -1 \\ 0 & 1 & 6 \end{bmatrix}
\]</div>
<ul>
<li>它是有效计算特征值的有用中间形式。先将矩阵化为海森堡形式可以使迭代算法收敛更快。</li>
</ul>
<h2 id="colab-notebook">编程练习（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">&para;</a></h2>
<ol>
<li>
<p>创建一个正交矩阵（旋转矩阵），乘以其转置，验证得到单位矩阵。尝试不同的角度。
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">theta</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">/</span> <span class="mi">4</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="n">Q</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">jnp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">),</span> <span class="o">-</span><span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)],</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="p">[</span><span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">),</span> <span class="n">jnp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)]])</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Q @ Q.T:</span><span class="se">\n</span><span class="si">{</span><span class="n">Q</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">Q</span><span class="o">.</span><span class="n">T</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Determinant: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>创建一个对称矩阵并验证它等于其转置。然后计算其特征值并检查特征向量是否垂直。
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="n">S</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">],</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a> <span class="p">[</span><span class="mf">2.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]])</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Symmetric: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">S</span><span class="p">,</span><span class="w"> </span><span class="n">S</span><span class="o">.</span><span class="n">T</span><span class="p">)</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">eigenvalues</span><span class="p">,</span> <span class="n">eigenvectors</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Eigenvalues: </span><span class="si">{</span><span class="n">eigenvalues</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Dot product of eigenvectors: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">eigenvectors</span><span class="p">[:,</span><span class="w"> </span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="n">eigenvectors</span><span class="p">[:,</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="si">:</span><span class="s2">.6f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
</ol>
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