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Sat, 11 Apr 2026 19:27:54 +0000 https://gb.yeadoc.com/matrix-decomposition/ Sat, 11 Apr 2026 19:27:54 +0000 https://gb.yeadoc.com/matrix-decomposition/ <p>特征向量-直方图</p> <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="ln">1</span><span class="cl"><span class="kn">from</span> <span class="nn">scipy.linalg</span> <span class="kn">import</span> <span class="n">eig</span> <span class="c1"># 矩阵分解</span> </span></span><span class="line"><span class="ln">2</span><span class="cl"><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span> <span class="c1"># 生成矩阵</span> </span></span><span class="line"><span class="ln">3</span><span class="cl"><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span> <span class="c1"># 绘图</span> </span></span></code></pre></div><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="ln">1</span><span class="cl"><span class="n">A</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span><span class="c1">#生成一个2*2的矩阵</span> </span></span><span class="line"><span class="ln">2</span><span class="cl"> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span> </span></span><span class="line"><span class="ln">3</span><span class="cl"><span class="n">A</span> </span></span></code></pre></div><p>[[1, 2], [2, 1]]</p> <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="ln">1</span><span class="cl"> </span></span><span class="line"><span class="ln">2</span><span class="cl"><span class="c1"># 特征分解</span> </span></span><span class="line"><span class="ln">3</span><span class="cl"><span class="n">evals</span><span class="p">,</span> <span class="n">evecs</span> <span class="o">=</span> <span class="n">eig</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="c1">#求A的特征值（evals）和特征向量(evecs)</span> </span></span></code></pre></div><p>[[ 0.70710678 -0.70710678] [ 0.70710678 0.70710678]]</p> <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="ln">1</span><span class="cl"><span class="n">evecs</span> <span class="o">=</span> <span class="n">evecs</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">evecs</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> </span></span><span class="line"><span class="ln">2</span><span class="cl"><span class="nb">print</span><span class="p">(</span><span class="n">evecs</span><span class="p">)</span> </span></span></code></pre></div><p>(array([0.70710678, 0.70710678]), array([-0.70710678, 0.70710678]))</p> <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="ln"> 1</span><span class="cl"><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span> </span></span><span class="line"><span class="ln"> 2</span><span class="cl"><span class="k">for</span> <span class="n">spine</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;left&#39;</span><span class="p">,</span> <span class="s1">&#39;bottom&#39;</span><span class="p">]:</span><span class="c1">#让在左下角的坐标轴经过原点</span> </span></span><span class="line"><span class="ln"> 3</span><span class="cl"> <span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="n">spine</span><span class="p">]</span><span class="o">.</span><span class="n">set_position</span><span class="p">(</span><span class="s1">&#39;zero&#39;</span><span class="p">)</span> </span></span><span class="line"><span class="ln"> 4</span><span class="cl"> <span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">)</span> </span></span><span class="line"><span class="ln"> 5</span><span class="cl"><span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span> </span></span><span class="line"><span class="ln"> 6</span><span class="cl"><span class="n">ymin</span><span class="p">,</span> <span class="n">ymax</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span> </span></span><span class="line"><span class="ln"> 7</span><span class="cl"><span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlim</span><span class="o">=</span><span class="p">(</span><span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">),</span> <span class="n">ylim</span><span class="o">=</span><span class="p">(</span><span class="n">ymin</span><span class="p">,</span> <span class="n">ymax</span><span class="p">))</span> </span></span><span class="line"><span class="ln"> 8</span><span class="cl"><span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">evecs</span><span class="p">:</span> </span></span><span class="line"><span class="ln"> 9</span><span class="cl"> <span class="n">ax</span><span class="o">.</span><span class="n">annotate</span><span class="p">(</span><span class="n">s</span><span class="o">=</span><span class="s2">&#34;&#34;</span><span class="p">,</span> <span class="n">xy</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">xytext</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> </span></span><span class="line"><span class="ln">10</span><span class="cl"> <span class="n">arrowprops</span><span class="o">=</span><span class="nb">dict</span><span class="p">(</span><span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;blue&#39;</span><span class="p">,</span> </span></span><span class="line"><span class="ln">11</span><span class="cl"> <span class="n">shrink</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> </span></span><span class="line"><span class="ln">12</span><span class="cl"> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.6</span><span class="p">,</span> </span></span><span class="line"><span class="ln">13</span><span class="cl"> <span class="n">width</span><span class="o">=</span><span class="mf">0.5</span><span class="p">))</span> </span></span><span class="line"><span class="ln">14</span><span class="cl"><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="c1">#在指定的间隔内返回均匀间隔的数字</span> </span></span><span class="line"><span class="ln">15</span><span class="cl"><span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">evecs</span><span class="p">:</span> </span></span><span class="line"><span class="ln">16</span><span class="cl"> <span class="n">a</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">/</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="c1">#沿特征向量方向的单位向量</span> </span></span><span class="line"><span class="ln">17</span><span class="cl"> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">a</span> <span class="o">*</span> <span class="n">x</span><span class="p">,</span> <span class="s1">&#39;r-&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mf">0.4</span><span class="p">)</span><span class="c1"># 参数 lw 表示图线的粗细</span> </span></span><span class="line"><span class="ln">18</span><span class="cl"><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span> </span></span></code></pre></div><p><figure> <picture> <img loading="lazy" decoding="async" alt="Image" class="image_figure image_internal image_unprocessed" src="https://gb.yeadoc.com/image.png" /> </picture> </figure> </p> <p>奇异值分解</p> <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="ln">1</span><span class="cl"><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span> </span></span><span class="line"><span class="ln">2</span><span class="cl"><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span> </span></span></code></pre></div><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="ln">1</span><span class="cl"><span class="n">words</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&#34;books&#34;</span><span class="p">,</span><span class="s2">&#34;dad&#34;</span><span class="p">,</span><span class="s2">&#34;stock&#34;</span><span class="p">,</span><span class="s2">&#34;value&#34;</span><span class="p">,</span><span class="s2">&#34;singular&#34;</span><span class="p">,</span><span class="s2">&#34;estate&#34;</span><span class="p">,</span><span class="s2">&#34;decomposition&#34;</span><span class="p">]</span> </span></span></code></pre></div><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="ln">1</span><span class="cl"><span class="n">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span> </span></span></code></pre></div><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="ln">1</span><span class="cl"><span class="n">X</span> </span></span></code></pre></div><p>array([[0, 2, 1, 0, 0, 0, 0], [2, 0, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 0, 1], [0, 1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 1, 1]])</p>