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<article class="md-content__inner md-typeset">
<h1 id="_1">矩阵性质<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>矩阵是存储数据集、编码变换和定义每个神经网络层的数据结构。本文涵盖矩阵维度、元素、转置、迹、行列式、逆、秩和零空间，这些是贯穿线性代数和 ML 的基础性质。</em></p>
<ul>
<li>核心而言，<strong>矩阵</strong>是按行列排列的数字矩形网格。如果向量是数字的单个列表，那么矩阵就是数字的一张表格。</li>
</ul>
<div class="arithmatex">\[
A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}
\]</div>
<ul>
<li>
<p>你也可以将矩阵视为向量的堆叠。</p>
</li>
<li>
<p>如果一个人由向量 <span class="arithmatex">\([\text{age}, \text{height}, \text{weight}]\)</span> 描述，那么三个人就形成一个矩阵，其中每行是一个人：</p>
</li>
</ul>
<div class="arithmatex">\[
\begin{bmatrix} 25 & 170 & 65 \\ 30 & 180 & 80 \\ 22 & 160 & 55 \end{bmatrix}
\]</div>
<ul>
<li>
<p>这个矩阵有 3 行和 3 列，所以我们称它为 <span class="arithmatex">\(3 \times 3\)</span> 矩阵。</p>
</li>
<li>
<p>网格中的每个数字称为一个<strong>元素</strong>或<strong>条目</strong>，由其行列标识：<span class="arithmatex">\(A_{ij}\)</span> 是第 <span class="arithmatex">\(i\)</span> 行第 <span class="arithmatex">\(j\)</span> 列的元素。</p>
</li>
<li>
<p>矩阵的<strong>转置</strong>沿其对角线翻转，将行变为列，列变为行。如果 <span class="arithmatex">\(A\)</span> 是 <span class="arithmatex">\(m \times n\)</span>，那么 <span class="arithmatex">\(A^T\)</span> 是 <span class="arithmatex">\(n \times m\)</span>。</p>
</li>
</ul>
<div class="arithmatex">\[
A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \quad \Rightarrow \quad A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}
\]</div>
<ul>
<li>
<p>矩阵乘以其转置总是得到一个方阵：<span class="arithmatex">\(AA^T\)</span> 是 <span class="arithmatex">\(m \times m\)</span>，<span class="arithmatex">\(A^TA\)</span> 是 <span class="arithmatex">\(n \times n\)</span>。</p>
</li>
<li>
<p>方阵的<strong>迹</strong>是其对角线元素之和：<span class="arithmatex">\(\text{tr}(A) = A_{11} + A_{22} + \cdots + A_{nn}\)</span>。迹等于特征值之和（我们稍后会看到）。</p>
</li>
</ul>
<p><img alt="迹：对角线元素之和" src="../../images/matrix_trace.svg" /></p>
<ul>
<li>
<p>对于上面的矩阵，<span class="arithmatex">\(\text{tr}(A) = 1 + 4 + 9 = 14\)</span>。只有高亮的对角线部分重要。</p>
</li>
<li>
<p>如果两个矩阵在不同基下表示相同的线性变换，它们的迹相同。迹是"与基无关的。"</p>
</li>
<li>
<p>矩阵的<strong>秩</strong>是线性无关的行（或等价地，列）的数量。它告诉你矩阵携带了多少"有用信息。"</p>
</li>
<li>
<p>例如，以下矩阵的秩为 2，因为两行之间互不为倍数：</p>
</li>
</ul>
<div class="arithmatex">\[
\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
\]</div>
<p>但以下矩阵的秩为 1，因为第二行只是第一行的两倍，所以它没有增加新信息：</p>
<div class="arithmatex">\[
\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}
\]</div>
<ul>
<li>一个 <span class="arithmatex">\(5 \times 3\)</span> 矩阵的秩最多为 3。如果某些行只是其他行的缩放或组合版本，秩就会下降。具有最大可能秩的矩阵称为<strong>满秩</strong>。</li>
</ul>
<p><img alt="秩：独立行张成整个空间 vs. 相关行只张成一个子空间" src="../../images/matrix_rank.svg" /></p>
<ul>
<li>
<p>方阵可逆（有逆矩阵）当且仅当它是满秩的。</p>
</li>
<li>
<p>秩通过<strong>秩-零化度定理</strong>与<strong>零空间</strong>（矩阵映射到零的向量的集合）相连：<span class="arithmatex">\(\text{rank}(A) + \text{nullity}(A) = \text{列数 of } A\)</span>。矩阵保留的（秩）加上它破坏的（零化度）等于总维度。</p>
</li>
<li>
<p>矩阵的<strong>列空间</strong>是当你将矩阵乘以任意向量时所有可能输出的集合。它由矩阵的列张成。如果矩阵有 3 列但只有 2 列独立，列空间是一个二维平面，而不是整个三维空间。</p>
</li>
</ul>
<p><img alt="列空间：独立列张成一个平面，相关列只张成一条线" src="../../images/column_space.svg" /></p>
<ul>
<li>
<p><strong>行空间</strong>是同样的概念，但从行的角度来看。秩等于列空间和行空间的维度，所以它们总是一致的。</p>
</li>
<li>
<p>一起来看，列空间告诉你"这个矩阵能产生什么输出？"零空间告诉你"什么输入被映射到零？"这两个空间完整描述了矩阵的功能。</p>
</li>
<li>
<p>方阵的<strong>行列式</strong>是一个标量，捕捉矩阵如何缩放空间。想象一个 <span class="arithmatex">\(2 \times 2\)</span> 矩阵将一个单位正方形变换成一个平行四边形。行列式就是那个平行四边形的面积（带有符号）。</p>
</li>
</ul>
<div class="arithmatex">\[
\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc
\]</div>
<p><img alt="行列式：线性变换的面积缩放因子" src="../../images/determinant.svg" /></p>
<ul>
<li>例如：</li>
</ul>
<div class="arithmatex">\[
\det\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} = 2 \cdot 3 - 1 \cdot 0 = 6
\]</div>
<p>这个变换将单位正方形拉伸成一个面积为 6 的平行四边形。</p>
<ul>
<li>
<p>如果行列式为正，变换保持定向（事物不会被"翻转"）。如果为负，它翻转定向（像镜面反射）。如果为零，矩阵将空间压缩到更低维度，将平行四边形坍缩成一条线或一个点。</p>
</li>
<li>
<p>行列式为零的矩阵称为<strong>奇异矩阵</strong>。它没有逆矩阵且已永久丢失信息。</p>
</li>
<li>
<p>对于大于 <span class="arithmatex">\(2 \times 2\)</span> 的矩阵，行列式使用<strong>余子式</strong>和<strong>代数余子式</strong>计算。<strong>余子式</strong> <span class="arithmatex">\(M_{ij}\)</span> 是通过删除第 <span class="arithmatex">\(i\)</span> 行和第 <span class="arithmatex">\(j\)</span> 列得到的较小矩阵的行列式。</p>
</li>
</ul>
<p><img alt="余子式：删除一行一列得到更小的矩阵" src="../../images/cofactor.svg" /></p>
<ul>
<li>
<p><strong>代数余子式</strong> <span class="arithmatex">\(C_{ij} = (-1)^{i+j} M_{ij}\)</span> 为每个余子式附加一个符号（像棋盘一样交替：<span class="arithmatex">\(+, -, +, \ldots\)</span>）。整个矩阵的行列式然后沿着任意行或列求和：<span class="arithmatex">\(\det(A) = \sum_j A_{1j} \cdot C_{1j}\)</span>。这称为<strong>代数余子式展开</strong>。</p>
</li>
<li>
<p>方阵 <span class="arithmatex">\(A\)</span> 的<strong>逆</strong>，记作 <span class="arithmatex">\(A^{-1}\)</span>，是撤销 <span class="arithmatex">\(A\)</span> 的矩阵：<span class="arithmatex">\(AA^{-1} = A^{-1}A = I\)</span>（单位矩阵）。只有非奇异矩阵才有逆。</p>
</li>
<li>
<p>对于 <span class="arithmatex">\(2 \times 2\)</span> 矩阵，逆有一个直接公式：</p>
</li>
</ul>
<div class="arithmatex">\[
\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]</div>
<p>注意分母中的行列式，这就是为什么奇异矩阵（行列式为零）没有逆。</p>
<ul>
<li>
<p><strong>条件数</strong>衡量矩阵对其输入微小变化的敏感程度。它定义为 <span class="arithmatex">\(\kappa(A) = \|A\| \cdot \|A^{-1}\|\)</span>。</p>
</li>
<li>
<p>接近 1 的条件数意味着矩阵是<strong>良态的</strong>：微小的输入变化产生微小的输出变化。大的条件数意味着它是<strong>病态的</strong>：微小的误差被极大放大。正交矩阵和单位矩阵的条件数为 1，而奇异矩阵的条件数为无穷大。</p>
</li>
<li>
<p>例如，以下矩阵的条件数为 <span class="arithmatex">\(10^8\)</span>。一个方向被正常缩放，而另一个几乎被压缩为零，所以沿该方向的小扰动会被严重扭曲：</p>
</li>
</ul>
<div class="arithmatex">\[
\begin{bmatrix} 1 & 0 \\ 0 & 10^{-8} \end{bmatrix}
\]</div>
<ul>
<li>就像向量有范数（长度）一样，矩阵也有衡量其"大小"的<strong>范数</strong>。最常见的是<strong>弗罗贝尼乌斯范数</strong>，它将矩阵视为一个长向量并计算其长度：</li>
</ul>
<div class="arithmatex">\[
\|A\|_F = \sqrt{\sum_{i}\sum_{j} A_{ij}^2}
\]</div>
<ul>
<li>例如：</li>
</ul>
<div class="arithmatex">\[
\left\|\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\right\|_F = \sqrt{1 + 4 + 9 + 16} = \sqrt{30} \approx 5.48
\]</div>
<ul>
<li>
<p><strong>谱范数</strong> <span class="arithmatex">\(\|A\|_2\)</span> 是 <span class="arithmatex">\(A\)</span> 的最大奇异值。它衡量矩阵可以拉伸任何单位向量的最大程度。在 ML 中，矩阵范数用于权重正则化（惩罚大权重）和监控训练稳定性。</p>
</li>
<li>
<p>对称矩阵 <span class="arithmatex">\(A\)</span> 是<strong>正定的</strong>，如果对每个非零向量 <span class="arithmatex">\(\mathbf{x}\)</span>：<span class="arithmatex">\(\mathbf{x}^T A \mathbf{x} &gt; 0\)</span>。这个二次型总是产生正数。</p>
</li>
<li>
<p>例如，以下矩阵是正定的：</p>
</li>
</ul>
<div class="arithmatex">\[
A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}
\]</div>
<p>取任意向量，比如 <span class="arithmatex">\(\mathbf{x} = [1, -1]^T\)</span>：<span class="arithmatex">\(\mathbf{x}^T A \mathbf{x} = 2 - 1 - 1 + 3 = 3 &gt; 0\)</span>。无论你尝试哪个非零 <span class="arithmatex">\(\mathbf{x}\)</span>，你总是得到正的结果。</p>
<ul>
<li>
<p>正定矩阵很重要，因为它们保证优化问题有唯一的最小值。</p>
</li>
<li>
<p>如果条件放宽到 <span class="arithmatex">\(\mathbf{x}^T A \mathbf{x} \geq 0\)</span>（允许为零），矩阵是<strong>半正定</strong>（PSD）。PSD 矩阵经常出现：协方差矩阵、SVM 中的核矩阵以及局部最小值处的 Hessian 矩阵都是 PSD。区别在于 PSD 允许某些方向是"平坦的"（零曲率），而不是严格向上弯曲。</p>
</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>计算矩阵的迹、秩和行列式。尝试使一行成为另一行的倍数，观察秩和行列式如何变化。
<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">A</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">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">],</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="p">[</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">4.0</span><span class="p">]])</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Trace: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">trace</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<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;Rank: </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">matrix_rank</span><span class="p">(</span><span class="n">A</span><span class="p">)</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">A</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>计算矩阵的逆，将其乘以原矩阵，验证得到单位矩阵。然后尝试奇异矩阵并观察会发生什么。
<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">A</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">1.0</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">3.0</span><span class="p">,</span> <span class="mf">4.0</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="n">A_inv</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">inv</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;A * A_inv:</span><span class="se">\n</span><span class="si">{</span><span class="n">A</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">A_inv</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
</ol>
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