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Book4_Ch13_Python_Codes/Bk4_Ch13_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_Ch13_01.py
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+A = np.array([[1.25,  -0.75],
+              [-0.75, 1.25]])
+
+xx1, xx2 = np.meshgrid(np.linspace(-8, 8, 9), np.linspace(-8, 8, 9))
+num_vecs = np.prod(xx1.shape);
+
+thetas = np.linspace(0, 2*np.pi, num_vecs)
+
+thetas = np.reshape(thetas, (-1, 9))
+thetas = np.flipud(thetas);
+
+uu = np.cos(thetas);
+vv = np.sin(thetas);
+
+fig, ax = plt.subplots()
+
+ax.quiver(xx1,xx2,uu,vv,
+          angles='xy', scale_units='xy',scale=1, 
+          edgecolor='none', facecolor= 'b')
+
+plt.ylabel('$x_2$')
+plt.xlabel('$x_1$')
+plt.axis('scaled')
+ax.set_xlim([-10, 10])
+ax.set_ylim([-10, 10])
+ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5])
+ax.set_xticks(np.linspace(-10,10,11));
+ax.set_yticks(np.linspace(-10,10,11));
+plt.show()
+
+# Matrix multiplication
+V = np.array([uu.flatten(),vv.flatten()]).T;
+W = V@A;
+
+uu_new = np.reshape(W[:,0],(-1, 9));
+vv_new = np.reshape(W[:,1],(-1, 9));
+
+fig, ax = plt.subplots()
+
+ax.quiver(xx1,xx2,uu,vv,
+          angles='xy', scale_units='xy',scale=1, 
+          edgecolor='none', facecolor= 'b')
+
+ax.quiver(xx1,xx2,uu_new,vv_new,
+          angles='xy', scale_units='xy',scale=1, 
+          edgecolor='none', facecolor= 'r')
+
+plt.ylabel('$x_2$')
+plt.xlabel('$x_1$')
+plt.axis('scaled')
+ax.set_xlim([-10, 10])
+ax.set_ylim([-10, 10])
+ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5])
+ax.set_xticks(np.linspace(-10,10,11));
+ax.set_yticks(np.linspace(-10,10,11));
+plt.show()
+
+
+fig, ax = plt.subplots()
+ax.quiver(xx1*0,xx2*0,uu,vv,
+          angles='xy', scale_units='xy',scale=1, 
+          edgecolor='none', facecolor= 'b')
+
+ax.quiver(xx1*0,xx2*0,uu_new,vv_new,
+          angles='xy', scale_units='xy',scale=1,
+          edgecolor='none', facecolor= 'r')
+
+plt.ylabel('$x_2$')
+plt.xlabel('$x_1$')
+plt.axis('scaled')
+ax.set_xlim([-2, 2])
+ax.set_ylim([-2, 2])
+ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5])
+ax.set_xticks(np.linspace(-2,2,5));
+ax.set_yticks(np.linspace(-2,2,5));
+plt.show()

Book4_Ch13_Python_Codes/Bk4_Ch13_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_Ch13_02.py
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+def visualize(X_circle,X_vec,title_txt):
+    
+    fig, ax = plt.subplots()
+    
+    plt.plot(X_circle[0,:], X_circle[1,:],'k',
+             linestyle = '--',
+             linewidth = 0.5)
+    
+    plt.quiver(0,0,X_vec[0,0],X_vec[1,0],
+              angles='xy', scale_units='xy',scale=1, 
+              color = [0, 0.4392, 0.7529])
+    
+    plt.quiver(0,0,X_vec[0,1],X_vec[1,1],
+              angles='xy', scale_units='xy',scale=1, 
+              color = [1,0,0])
+    
+    plt.axvline(x=0, color= 'k', zorder=0)
+    plt.axhline(y=0, color= 'k', zorder=0)
+    
+    plt.ylabel('$x_2$')
+    plt.xlabel('$x_1$')
+    
+    ax.set_aspect(1)
+    ax.set_xlim([-2.5, 2.5])
+    ax.set_ylim([-2.5, 2.5])
+    ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5])
+    ax.set_xticks(np.linspace(-2,2,5));
+    ax.set_yticks(np.linspace(-2,2,5));
+    plt.title(title_txt)
+    plt.show()
+
+
+theta = np.linspace(0, 2*np.pi, 100)
+
+circle_x1 = np.cos(theta)
+circle_x2 = np.sin(theta)
+
+V_vec = np.array([[np.sqrt(2)/2, -np.sqrt(2)/2],
+                  [np.sqrt(2)/2,  np.sqrt(2)/2]])
+
+X_circle = np.array([circle_x1, circle_x2])
+
+# plot original circle and two vectors
+visualize(X_circle,V_vec,'Original')
+
+A = np.array([[1.25, -0.75],
+              [-0.75, 1.25]])
+
+# plot the transformation of A
+
+visualize(A@X_circle, A@V_vec,'$A$')
+
+
+#%% Eigen deomposition
+
+# A = V @ D @ V.T
+
+lambdas, V = np.linalg.eig(A)
+
+D = np.diag(np.flip(lambdas))
+V = V.T # reverse the order
+
+print('=== LAMBDA ===')
+print(D)
+print('=== V ===')
+print(V)
+
+# plot the transformation of V.T
+
+visualize(V.T@X_circle, V.T@V_vec,'$V^T$')
+
+# plot the transformation of D @ V.T
+
+visualize(D@V.T@X_circle, D@V.T@V_vec,'$\u039BV^T$')
+
+# plot the transformation of V @ D @ V.T
+
+visualize(V@D@V.T@X_circle, V@D@V.T@V_vec,'$V\u039BV^T$')
+
+# plot the transformation of A
+
+visualize(A@X_circle, A@V_vec,'$A$')
+