New file for lecture

import numpy as np
A = np.array([[5,1],[1,2]])
print(A)

[[5 1]
 [1 2]]

import matplotlib.pyplot as plt

theta = np.linspace(0,2*np.pi, 100, endpoint=False)
xx = np.cos(theta)
yy = np.sin(theta)
plt.plot(xx,yy,'.')

[<matplotlib.lines.Line2D at 0x7fe655b197f0>]

X = np.concatenate((xx.reshape((1,100)),yy.reshape((1,100))),axis=0)
print(X[:,0:5])

[[1.         0.99802673 0.9921147  0.98228725 0.96858316]
 [0.         0.06279052 0.12533323 0.18738131 0.24868989]]

Y = np.matmul(A, X)
plt.plot(Y[0,:],Y[1,:],'.')

[<matplotlib.lines.Line2D at 0x7fe655ea7af0>]

print(np.linalg.det(A))

8.999999999999998

Lambda,V = np.linalg.eig(A)
print(Lambda)
print(V)

[5.30277564 1.69722436]
[[ 0.95709203 -0.28978415]
 [ 0.28978415  0.95709203]]

plt.plot([0,V[0,0]],[0,V[1,0]])
plt.plot([0,V[0,1]],[0,V[1,1]])

[<matplotlib.lines.Line2D at 0x7fe6564dddf0>]

plt.plot([0,Lambda[0]*V[0,0]],[0,Lambda[0]*V[1,0]])
plt.plot([0,Lambda[1]*V[0,1]],[0,Lambda[1]*V[1,1]])
plt.plot(Y[0,:],Y[1,:],'.')

[<matplotlib.lines.Line2D at 0x7fe6566c5fd0>]

print(np.matmul(A,V[:,0]))
print(Lambda[0]*V[:,0])

[5.07524428 1.53666032]
[5.07524428 1.53666032]