diff --git a/Book4_Ch12_Cholesky分解__数学要素__从加减乘除到机器学习.pdf b/Book4_Ch12_Cholesky分解__数学要素__从加减乘除到机器学习.pdf
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diff --git a/Book4_Ch12_Python_Codes/Bk4_Ch12_01.py b/Book4_Ch12_Python_Codes/Bk4_Ch12_01.py
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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch12_01.py
+
+import numpy as np
+from matplotlib import pyplot as plt
+
+x1 = np.arange(-2,2,0.05)
+x2 = np.arange(-2,2,0.05)
+
+xx1_fine, xx2_fine = np.meshgrid(x1,x2)
+
+a = 0; b = -1; c = 0;
+
+yy_fine = a*xx1_fine**2 + 2*b*xx1_fine*xx2_fine + c*xx2_fine**2
+
+# 3D visualization
+
+fig, ax = plt.subplots()
+ax = plt.axes(projection='3d')
+
+ax.plot_wireframe(xx1_fine,xx2_fine,yy_fine,
+                  color = [0.8,0.8,0.8],
+                  linewidth = 0.25)
+
+ax.contour3D(xx1_fine,xx2_fine,yy_fine,15,
+             cmap = 'RdYlBu_r')
+
+ax.view_init(elev=30, azim=60)
+ax.xaxis.set_ticks([])
+ax.yaxis.set_ticks([])
+ax.zaxis.set_ticks([])
+plt.tight_layout()
+ax.set_proj_type('ortho')
+plt.show()
+
+# 2D visualization
+
+fig, ax = plt.subplots()
+
+ax.contourf(xx1_fine,xx2_fine,yy_fine,15,
+           cmap = 'RdYlBu_r')
+
+ax.xaxis.set_ticks([])
+ax.yaxis.set_ticks([])
+ax.set_aspect('equal')
+
+plt.show()
diff --git a/Book4_Ch12_Python_Codes/Bk4_Ch12_02.py b/Book4_Ch12_Python_Codes/Bk4_Ch12_02.py
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+++ b/Book4_Ch12_Python_Codes/Bk4_Ch12_02.py
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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch12_02.py
+
+import matplotlib.pyplot as plt
+from mpl_toolkits.mplot3d import Axes3D
+import numpy as np
+import math as m
+
+cos_theta_12 = np.cos(m.radians(135))
+cos_theta_13 = np.cos(m.radians(60))
+cos_theta_23 = np.cos(m.radians(120))
+
+P = np.array([[1, cos_theta_12, cos_theta_13],
+              [cos_theta_12, 1, cos_theta_23],
+              [cos_theta_13, cos_theta_23, 1]])
+
+L = np.linalg.cholesky(P)
+R = L.T
+
+
+fig = plt.figure()
+ax = fig.add_subplot(projection='3d')
+
+plt.plot(0,0,0,color = 'r', marker = 'x',
+         markersize = 12)
+
+colors = ['b', 'r', 'g']
+for i in np.arange(0,3):
+    
+    vector = R[:,i]
+    v = np.array([vector[0],vector[1],vector[2]])
+    vlength=np.linalg.norm(v)
+    ax.quiver(0,0,0,vector[0],vector[1],vector[2],
+            length=vlength, color = colors[i])
+
+ax.set_xlim([-1,1])
+ax.set_ylim([-1,1])
+ax.set_zlim([-1,1])
+
+ax.set_xlabel('x')
+ax.set_ylabel('y')
+ax.set_zlabel('z')
+
+ax.xaxis.set_ticklabels([])
+ax.yaxis.set_ticklabels([])
+ax.zaxis.set_ticklabels([])
+ax.view_init(35, 60)
+ax.set_proj_type('ortho')
+ax.set_box_aspect([1,1,1])