In [1]:

    

from scipy import linalg

import numpy as np


    

In [2]:

    

A = np.array([[1, 2], [3, 4]])


    

In [3]:

    

# Matrix Multiplication
A.dot(A)


    

    
Out[3]:





array([[ 7, 10],
       [15, 22]])

In [4]:

    

# Matrix Norm
linalg.norm(A)


    

    
Out[4]:





5.477225575051661

In [5]:

    

# CHECK: Matrix Norm
np.sqrt(sum([x**2 for x in range(1, 5)]))


    

    
Out[5]:





5.477225575051661

In [8]:

    

# Calc determinant
# For 2x2 matrix is a*d - b*c
linalg.det(A)


    

    
Out[8]:





-2.0

In [9]:

    

print(A)


    

    




[[1 2]
 [3 4]]

In [10]:

    

# Calc Matrix Inverse
A_inv = linalg.inv(A)


    

In [11]:

    

print(A_inv)


    

    




[[-2.   1. ]
 [ 1.5 -0.5]]

In [12]:

    

# CHECK: Inverse X original = identity matrix
A_inv.dot(A)


    

    
Out[12]:





array([[1.00000000e+00, 0.00000000e+00],
       [1.11022302e-16, 1.00000000e+00]])

In [13]:

    

# Compute Eigenvalues


    

In [14]:

    

# Decomposes A into a
# diagonal matrix x and invertible matrix y such that y * x * y^-1
x, y = linalg.eig(A)

x0, x1 = x[0], x[1]

y0, y1 = y[:, 0], y[:, 1]


    

In [25]:

    

y


    

    
Out[25]:





array([[-0.82456484, -0.41597356],
       [ 0.56576746, -0.90937671]])

In [26]:

    

# Verify normal equation #1
print(A.dot(y0))


    

    




[ 0.30697009 -0.21062466]

In [27]:

    

print(x0 * y0)


    

    




[ 0.30697009-0.j -0.21062466+0.j]

In [28]:

    

# Verify normal equation #2

print(A.dot(y1))


    

    




[-2.23472698 -4.88542751]

In [29]:

    

print(x1 * y1)


    

    




[-2.23472698+0.j -4.88542751+0.j]

In [30]:

    

# verify eigenvalue decomposition
y.dot(np.diag(x).dot(linalg.inv(y)))


    

    
Out[30]:





array([[1.+0.j, 2.+0.j],
       [3.+0.j, 4.+0.j]])

In [31]:

    

# Calc singular value decomposition
# Extension to eigenvalues for non-invertible or non-square matrices
# product of an orthogonal matrix * diagonal matrix * another orthogonal matrix
u, s, v = linalg.svd(A)


    

In [32]:

    

# U is symmetric & orthogonal
print(u)

print(u.dot(u.T))


    

    




[[-0.40455358 -0.9145143 ]
 [-0.9145143   0.40455358]]
[[1. 0.]
 [0. 1.]]

In [18]:

    

# Sigma is a diagonal matrix (if A is invertible)
print(np.diag(s))


    

    




[[5.4649857  0.        ]
 [0.         0.36596619]]

In [19]:

    

# V is symmetric
print(v)

# V is orthogonal
print(v.dot(v.T))


    

    




[[-0.57604844 -0.81741556]
 [ 0.81741556 -0.57604844]]
[[1. 0.]
 [0. 1.]]

In [20]:

    

# CHECK: singular value decomposition recovers A
u.dot(np.diag(s).dot(v))


    

    
Out[20]:





array([[1., 2.],
       [3., 4.]])

In [ ]: