diff --git a/素材/整合素材/正交及二次型.md b/素材/整合素材/正交及二次型.md
index 8cfadb8..1f4c336 100644
--- a/素材/整合素材/正交及二次型.md
+++ b/素材/整合素材/正交及二次型.md
@@ -116,15 +116,9 @@ $$\|\boldsymbol{\gamma}_3\|=\sqrt{\dfrac{3}{2}},\boldsymbol{\varepsilon}_3=\dfra
 >解上述齐次方程组,得到两个线性无关的解：
 >$$\boldsymbol{\xi}_1=(2,1,4,0)^T,\quad
 \boldsymbol{\xi}_2=(0,1,-1,1)^T$$
->正交化
->$$\boldsymbol{\beta}_1 = \boldsymbol{\xi}_1 = (2,1,4,0)^T$$
->$$\boldsymbol{\beta}_2 = \boldsymbol{\xi}_2 - \frac{\langle\boldsymbol{\xi}_2,\boldsymbol{\beta}_1\rangle}{\langle\boldsymbol{\beta}_1,\boldsymbol{\beta}_1\rangle}\boldsymbol{\beta}_1
-= (0,1,0,1)^T - \frac{1}{21}(2,1,4,0)^T
-= \left(-\frac{2}{21},\frac{20}{21},-\frac{4}{21},1\right)^T$$
->单位化$$\boldsymbol{\alpha}_1 = \frac{\boldsymbol{\beta}_1}{\|\boldsymbol{\beta}_1\|}=\frac{1}{\sqrt{21}}(2,1,4,0)^T$$
->$$\boldsymbol{\alpha}_2 = \frac{\boldsymbol{\beta}_2}{\|\boldsymbol{\beta}_2\|}
-=\frac{1}{3\sqrt{105}}(-2,20,-4,21)^T$$
->满足条件的一组标准正交向量为：$$\boldsymbol{\alpha}_1 = \frac{1}{\sqrt{21}}\begin{bmatrix}2\\1\\4\\0\end{bmatrix},\quad\boldsymbol{\alpha}_2 = \frac{1}{3\sqrt{105}}\begin{bmatrix}-2\\20\\-4\\21\end{bmatrix}$$
+>用施密特正交化法得 $$\boldsymbol u_1=\boldsymbol \xi_1,\boldsymbol\varepsilon_1=\dfrac{\boldsymbol u_1}{\|\boldsymbol u_1\|}=\dfrac{1}{\sqrt{21}}(2,1,4,0)^\text{T},$$
+>$$\boldsymbol u_2=\boldsymbol \xi_2-\langle\boldsymbol \xi_2,\boldsymbol \varepsilon_1\rangle\boldsymbol \varepsilon_1=\dfrac{1}{7}(2,8,-3,7)^\text{T},\boldsymbol \varepsilon_2=\dfrac{\boldsymbol u_2}{\|\boldsymbol u_2\|}=\dfrac{1}{4\sqrt 5}(2,0,-3,7)^\text T$$
+>满足条件的一组标准正交向量为：$$\boldsymbol{\varepsilon}_1 = \frac{1}{\sqrt{21}}\begin{bmatrix}2\\1\\4\\0\end{bmatrix},\quad\boldsymbol{\varepsilon}_2 = \frac{1}{4\sqrt{5}}\begin{bmatrix}2\\0\\-3\\7\end{bmatrix}.$$
 
 # Section 2  实对称矩阵的正交变换与二次型
 ## 实对称矩阵在相似变换时的特殊性