月更 技术博客 CXX优先 Zola 2026-08-11T00:00:00+00:00 https://suchaharcan.github.io/tags/linearalgebra/atom.xml 2026-08-11T00:00:00+00:00 2026-08-11T00:00:00+00:00 Suchaharcan https://suchaharcan.github.io/00Read_blog_Math/math-linear-algebra/ <h1 id="xian-xing-dai-shu-he-xin-zhang-jie">线性代数 · 核心章节</h1> <blockquote> <p>线代脉络：<strong>行列式 → 矩阵 → 向量组 → 方程组 → 特征值 → 二次型</strong>，前四章是后两章的工具。</p> </blockquote> <hr /> <h2 id="1-xing-lie-shi">1. 行列式</h2> <h3 id="ding-yi-yu-xing-zhi">定义与性质</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>det(A) = Σ σ(p) a₁p₁ a₂p₂ … aₙpₙ</span></span></code></pre> <p>核心性质：</p> <ul> <li><code>|Aᵀ| = |A|</code></li> <li>两行（列）互换，行列式变号；</li> <li>某行乘以 k，行列式乘以 k；</li> <li>两行成比例 ⇒ 行列式为 0；</li> <li>某行加另一行的 k 倍，值不变；</li> <li><code>|AB| = |A|·|B|</code>；<code>|A⁻¹| = 1/|A|</code>；<code>|kA| = kⁿ|A|</code></li> </ul> <h3 id="zhan-kai-ding-li-dai-shu-yu-zi-shi">展开定理（代数余子式）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>按第 i 行展开：|A| = Σ(j) aᵢⱼ·Aᵢⱼ</span></span></code></pre><h3 id="te-shu-xing-lie-shi">特殊行列式</h3> <table><thead><tr><th>类型</th><th>结果</th></tr></thead><tbody> <tr><td>上/下三角</td><td>主对角线乘积</td></tr> <tr><td>对角</td><td>对角线乘积</td></tr> <tr><td>范德蒙德</td><td>∏(xⱼ−xᵢ)（i&lt;j）</td></tr> <tr><td>副对角</td><td><code>(−1)^(n(n−1)/2)·∏aᵢ,ₙ₋ᵢ₊₁</code></td></tr> </tbody></table> <h3 id="ke-la-mo-fa-ze">克拉默法则</h3> <p><code>Ax=b</code> 且 <code>|A|≠0</code> 时：<code>xᵢ = |Aᵢ|/|A|</code>（Aᵢ 为替换第 i 列）。</p> <hr /> <h2 id="2-ju-zhen">2. 矩阵</h2> <h3 id="yun-suan">运算</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>(AB)C = A(BC)；A(B+C) = AB + AC</span></span> <span class="giallo-l"><span>(AB)ᵀ = BᵀAᵀ；(A+B)ᵀ = Aᵀ+Bᵀ</span></span> <span class="giallo-l"><span>|AB| = |A||B|</span></span></code></pre><h3 id="ni-ju-zhen">逆矩阵</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>A⁻¹ 存在 ⇔ |A| ≠ 0 ⇔ A 可逆 ⇔ r(A) = n</span></span> <span class="giallo-l"><span>A⁻¹ = A*/|A|（伴随矩阵法）</span></span> <span class="giallo-l"><span>(AB)⁻¹ = B⁻¹A⁻¹；(Aᵀ)⁻¹ = (A⁻¹)ᵀ</span></span></code></pre> <p>常用公式：<code>A·A* = |A|E</code> ⇒ <code>A* = |A|A⁻¹</code>。</p> <h3 id="chu-deng-bian-huan-yu-chu-deng-ju-zhen">初等变换与初等矩阵</h3> <p>初等行变换不改变秩；初等矩阵左乘 = 行变换，右乘 = 列变换。</p> <h3 id="zhi">秩</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>r(A)：非零子式的最高阶数</span></span> <span class="giallo-l"><span>r(A) ≤ min(m,n)；r(A+B) ≤ r(A)+r(B)</span></span> <span class="giallo-l"><span>r(AB) ≤ min(r(A), r(B))；r(A)+r(B) − n ≤ r(AB)</span></span> <span class="giallo-l"><span>r(A)=r(Aᵀ)；r(kA)=r(A)（k≠0）</span></span></code></pre> <hr /> <h2 id="3-xiang-liang-zu">3. 向量组</h2> <h3 id="xian-xing-xiang-guan-wu-guan">线性相关 / 无关</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>α₁,…,αₛ 线性相关 ⇔ 存在不全为 0 的 kᵢ 使 Σ kᵢαᵢ = 0</span></span> <span class="giallo-l"><span>无关 ⇔ 只有全 0 系数成立</span></span></code></pre> <p>判定：</p> <ul> <li><code>s &gt; n</code>（n 维向量）必相关；</li> <li>含零向量 ⇒ 相关；</li> <li>部分相关 ⇒ 整体相关；整体无关 ⇒ 部分无关；</li> <li>等价向量组秩相同。</li> </ul> <h3 id="zhi-yu-ji-da-wu-guan-zu">秩与极大无关组</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>秩 = 极大线性无关组中向量个数</span></span> <span class="giallo-l"><span>向量组秩 ≤ 矩阵秩，且 r(α₁,…,αₛ) = r(矩阵)</span></span></code></pre> <hr /> <h2 id="4-xian-xing-fang-cheng-zu">4. 线性方程组</h2> <p><code>Ax = b</code>（m×n）</p> <h3 id="jie-de-cun-zai-xing">解的存在性</h3> <table><thead><tr><th>条件</th><th>结论</th></tr></thead><tbody> <tr><td>`r(A) &lt; r(A</td><td>b)`</td></tr> <tr><td>`r(A) = r(A</td><td>b) = n`</td></tr> <tr><td>`r(A) = r(A</td><td>b) = r &lt; n`</td></tr> </tbody></table> <p>齐次 <code>Ax=0</code>：非零解 ⇔ <code>r(A) &lt; n</code> ⇔ <code>|A|=0</code>（方阵）。</p> <h3 id="jie-de-jie-gou">解的结构</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>非齐次通解 = 齐次通解 + 一个特解</span></span> <span class="giallo-l"><span>齐次通解 = C₁ξ₁ + … + Cₙ₋ᵣξₙ₋ᵣ（基础解系）</span></span></code></pre> <hr /> <h2 id="5-te-zheng-zhi-yu-te-zheng-xiang-liang">5. 特征值与特征向量</h2> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>Ax = λx（x ≠ 0）</span></span> <span class="giallo-l"><span>特征多项式：|λE − A| = 0</span></span></code></pre><h3 id="zhong-yao-xing-zhi">重要性质</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>Σλᵢ = tr(A)（迹）</span></span> <span class="giallo-l"><span>∏λᵢ = |A|</span></span> <span class="giallo-l"><span>不同特征值对应的特征向量线性无关</span></span> <span class="giallo-l"><span>A⁻¹ 特征值 1/λ；Aᵀ 与 A 特征值相同；A^k 特征值 λ^k</span></span></code></pre><h3 id="xiang-si-dui-jiao-hua">相似对角化</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>A ~ Λ ⇔ A 有 n 个线性无关特征向量 ⇔ 每个 k 重特征值对应 k 个线性无关特征向量</span></span> <span class="giallo-l"><span>P⁻¹AP = diag(λ₁,…,λₙ)，P 的列 = 特征向量</span></span> <span class="giallo-l"><span>Aⁿ = P·Λⁿ·P⁻¹</span></span></code></pre> <hr /> <h2 id="6-er-ci-xing">6. 二次型</h2> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>f = xᵀAx（A 对称）</span></span></code></pre><h3 id="biao-zhun-xing-yu-gui-fan-xing">标准形与规范形</h3> <ul> <li>配方法 / 正交变换法化标准形；</li> <li>规范形：系数只有 <code>1, −1, 0</code>；</li> <li><strong>惯性定理</strong>：正、负惯性指数唯一；</li> <li><strong>合同</strong>：<code>CᵀAC = B</code>（C 可逆）⇔ 正负惯性指数相同。</li> </ul> <h3 id="zheng-ding-pan-ding">正定判定</h3> <table><thead><tr><th>方法</th><th>条件</th></tr></thead><tbody> <tr><td>特征值</td><td>全为正</td></tr> <tr><td>顺序主子式</td><td>全大于 0</td></tr> <tr><td>定义</td><td><code>∀x≠0：xᵀAx &gt; 0</code></td></tr> </tbody></table> <hr /> <h2 id="7-kao-yan-chang-kao-ti-xing">7. 考研常考题型</h2> <ol> <li>行列式性质与抽象行列式计算；</li> <li>求逆、伴随矩阵、<code>A*</code> 与 <code>A</code> 的关系；</li> <li>秩的不等式；</li> <li>向量组相关性的参数讨论；</li> <li>含参数方程组的解判定与通解；</li> <li>特征值特征向量、相似对角化、<code>Aⁿ</code>；</li> <li>二次型化标准形、正定判断。</li> </ol> <h3 id="li-ti-su-lan">例题速览</h3> <p><strong>例 1</strong>：已知 <code>A</code> 三阶，<code>|A|=2</code>，求 <code>|3A⁻¹|</code>。</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>|3A⁻¹| = 3³·|A⁻¹| = 27·(1/2) = 27/2</span></span></code></pre> <p><strong>例 2</strong>：<code>A</code> 特征值 1, 2, 3，求 <code>|A²+A+E|</code>。</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>A 的特征多项式作用于 A：g(λ) = λ²+λ+1</span></span> <span class="giallo-l"><span>|g(A)| = g(1)·g(2)·g(3) = 3·7·13 = 273</span></span></code></pre> <hr /> <h2 id="8-yi-cuo-dian-qing-dan">8. 易错点清单</h2> <ul> <li><code>(AB)ᵀ = BᵀAᵀ</code> 次序互换；</li> <li><code>|kA| = kⁿ|A|</code> 不是 <code>k|A|</code>；</li> <li>矩阵乘法不满足交换律；</li> <li><code>AB=0</code> 推不出 <code>A=0</code> 或 <code>B=0</code>；</li> <li>相似（存在可逆 P）与合同（对称阵）别混淆；</li> <li>特征向量必须非零。</li> </ul> <blockquote> <p>上一篇：<a href="https://suchaharcan.github.io/00Read_blog_Math/math-calculus-multi/">多元微积分</a> ｜ 下一篇：<a href="https://suchaharcan.github.io/00Read_blog_Math/math-probability/">概率论</a></p> </blockquote>