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Larry0454 · Sep 4, 2024 · 43e1cec · 43e1cec
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diff --git a/2024/09/01/linear_algebra/inner-product-space/index.html b/2024/09/01/linear_algebra/inner-product-space/index.html
@@ -24,7 +24,7 @@
 <meta property="og:description" content="内积空间  本篇是对高等代数中&quot;内积空间&quot;的定义、性质、定理以及计算方法的速记，不涉及严密的推导证明。  一、\(n\) 维内积与欧氏空间  内积定义：设 \(\alpha&#x3D;(a_1, \dots, a_n)^T, \beta &#x3D; (b_1, \dots, b_n)^T \in \mathbb{R}^n\)，定义内积 \((\alpha, \beta) &#x3D; \alpha^T \be">
 <meta property="og:locale" content="en_US">
 <meta property="article:published_time" content="2024-09-01T15:23:40.000Z">
-<meta property="article:modified_time" content="2024-09-04T14:45:19.473Z">
+<meta property="article:modified_time" content="2024-09-04T14:46:16.845Z">
 <meta property="article:author" content="WangLe">
 <meta property="article:tag" content="inner product">
 <meta property="article:tag" content="Schmidt orthogonalization">

diff --git a/2024/09/04/linear_algebra/quadratic-form/index.html b/2024/09/04/linear_algebra/quadratic-form/index.html
@@ -24,7 +24,7 @@
 <meta property="og:description" content="二次型  本篇是对高等代数中&quot;二次型&quot;的定义、定理以及计算方法的速记，不涉及严密的推导证明。  一、二次型的矩阵表示  \(n\) 元变量的二次齐次多项式：\(f(x_1, \dots, x_n) &#x3D; \sum_{i&#x3D;1}^n\sum_{j&#x3D;1}^n a_{ij}x_ix_j\)，其中 \(a_{ij} &#x3D; a_{ji}\) 二次型对应的矩阵：由 \(a_{ij} &#x3D; a_{">
 <meta property="og:locale" content="en_US">
 <meta property="article:published_time" content="2024-09-04T14:39:30.000Z">
-<meta property="article:modified_time" content="2024-09-04T14:44:19.897Z">
+<meta property="article:modified_time" content="2024-09-04T14:51:12.189Z">
 <meta property="article:author" content="WangLe">
 <meta property="article:tag" content="diagonalization">
 <meta property="article:tag" content="quadratic form">
@@ -380,8 +380,8 @@ <h3 id="三惯性定理">三、惯性定理</h3>
 <strong><u>负平方项数量</u></strong> <span class="math inline">\(q\)</span> 都是<strong>不变</strong>的</p></li>
 <li><p>合同标准型：实对称矩阵 <span class="math inline">\(\text{A}
 \simeq \text{diag}(1, \dots, 1; -1, \dots, -1;
-\textbf{O})\)</span>，其中 <span class="math inline">\(1\)</span> 有
-<span class="math inline">\(p\)</span> 个，<span class="math inline">\(-1\)</span> 有 <span class="math inline">\(q\)</span> 个，称为 <span class="math inline">\(\text{A}\)</span>
+\textbf{O})\)</span>，一共 <span class="math inline">\(p\)</span> 个
+<span class="math inline">\(1\)</span>，<span class="math inline">\(q\)</span> 个 <span class="math inline">\(-1\)</span>，称为 <span class="math inline">\(\text{A}\)</span>
 的<strong>合同标准型</strong></p>
 <p>标准型：合同规范型对应的<u>二次方程</u> <span class="math inline">\(y_1^2 + \dots + y_p^2 - y_{p+1}^2 - \dots-
 y_{p+q}^2\)</span> 称为 <span class="math inline">\(x^T\text{A}x\)</span>
@@ -466,7 +466,8 @@ <h3 id="三惯性定理">三、惯性定理</h3>
 <li>二次型 <span class="math inline">\(x^T\text{A}x\)</span>
 <u>半负定</u> <span class="math inline">\(\Leftrightarrow\)</span>
 <u>各阶</u>顺序主子式 <span class="math inline">\(\det (\text{A}_k) \le
-0\)</span>，且 $ (_i) = 0 $</li>
+0\)</span>，且 <span class="math inline">\(\exists \ \det({\text{A}_i})
+= 0\)</span></li>
 </ul></li>
 </ul>