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@alyssaq
Last active Aug 29, 2015
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Visualising eigenvectors and eigenvalues for the post http://scriptogr.am/alyssa/post/understanding-eigenvectors-and-eigenvalues-visually
import numpy as np
import matplotlib.pyplot as plt
def plot_points(matrix, ls='--', lw=1.2, colors=None):
"""
Plots a 2xn matrix where 1st row are the x-coordinates
and 2nd row are the y-coordinates.
Parameters:
matrix - 2xn matrix
ls - matplotlib linestyle
lw - matplotlib linewidth
colors - 1D array of colours (http://matplotlib.org/examples/color/named_colors.html)
"""
x_points, y_points = matrix
size = len(x_points)
colors = ['red', 'blue', 'orange', 'green'] if not None else colors
for i in range(size):
plt.plot(x_points[i], y_points[i], color=colors[i], marker='o')
plt.plot([x_points[i], x_points[(i+1) % size]],
[y_points[i], y_points[(i+1) % size]],
color=colors[i], linestyle=ls, linewidth=lw)
def plot_point_label(prefix, point, location):
"""
Plots a (x, y) point as a label
Parameters:
prefix - The name given to the point. E.g. 'p1'
point - 1D array of [x, y]
location - tuple coordinates of the label
"""
plt.annotate('{0}={1}'.format(prefix, tuple(map(lambda x: round(x, 2), point))),
xy=point, xytext=location, textcoords='data', color='white', weight='semibold',
bbox=dict(fc='navy', alpha=0.6, ec='none'), arrowprops=dict(arrowstyle='->'))
A = np.matrix([[1, 0.3], [0.45, 1.2]])
transformed_matrix = A * matrix
plot_points(matrix)
plot_points(transformed_matrix.A, '-', lw=3.0)
evals, evecs = np.linalg.eig(A)
x_v1, y_v1 = evecs[:,0].getA1()
x_v2, y_v2 = evecs[:,1].getA1()
m1 = y_v1/x_v1 # Gradient of 1st eigenvector
m2 = y_v2/x_v2 # Gradient of 2nd eigenvector
p1 = [-10/m1, -10] # 1st point at y = -10
p2 = [20/m2, 20] # 2nd point at y = -20
trans_p1 = A*np.matrix(p1).T
trans_p2 = A*np.matrix(p2).T
# Plot eigenvectors and labels
plt.plot([x_v1*-50, x_v1*50], [y_v1*-50, y_v1*50], color='royalblue')
plt.plot([x_v2*-50, x_v2*50], [y_v2*-50, y_v2*50], color='crimson')
plt.annotate('e1', xy=(-20, 14),textcoords='data', weight='semibold', color='royalblue')
plt.annotate('e2', xy=(-11, -20),textcoords='data', weight='semibold', color='crimson')
# Plot the points where the eigenvector line and original points intersect
plt.plot(p1[0], p1[1], 'ko')
plt.plot(p2[0], p2[1], 'ko')
# Plot the transformed points that lie on the eigenvector line
plt.plot(trans_p1[0,0], trans_p1[1,0], 'ko')
plt.plot(trans_p2[0,0], trans_p2[1,0], 'ko')
# Plot the point labels
plot_point_label('p1', p1, (12, -19))
plot_point_label('T(p1)', trans_p1.A1, (18, -13))
plot_point_label('p2', p2, (-14, 26))
plot_point_label('T(p2)', trans_p2.A1, (-6, 33))
# Set the axes
ax = plt.axes()
ax.spines['left'].set_position('zero')
ax.spines['right'].set_color('none')
ax.spines['bottom'].set_position('zero')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.set_aspect('equal')
# Limit the plot
plt.xlim([-20, 30])
plt.ylim([-20, 35])
plt.show()
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