import numpy as np

n = 10000

x1 = np.random.normal(0,1,n)
x2 = np.random.normal(0,1,n)
x3 = x1+x2
X = [x1,x2,x3]

c = np.cov(X)
print(c)
# Covariance matrix:
# [[ 1.00552305 -0.00540942  1.00011363]
#  [-0.00540942  1.00327819  0.99786877]
#  [ 1.00011363  0.99786877  1.9979824 ]]

e = np.linalg.eig(c)
print(e[0])
# Eigenvalues:
# [  2.99277382e+00   1.00136031e+00   1.76529821e-15]

print(e[1])
# Eigenvectors
# [[-0.40445603  0.70928273  0.57735027]
#  [-0.41202885 -0.70491056  0.57735027]
#  [-0.81648487  0.00437217 -0.57735027]]

# Cov(y) = M * Cov(x) * M^T
cy = np.dot(np.dot(e[1].T,c), e[1].T)
print(cy)
#  
# [[ 2.61908171  0.80509741 -0.50471862]
#  [ 0.80509741  0.98241759  0.30643185]
#  [-0.50471862  0.30643185  0.38716296]]