import numpy as np
import numpy.linalg as npl

# Use double brackets to ensure proper vector representation
x = np.array([1,2,3,4])
print(x)
print(x.T)
z = np.array([[1,2,3,4]])
print(z)
print(z.T)

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

# NumPy will do what it thinks you want it to...
# works...
npl.norm(x)**2
np.dot(x, x)

npl.norm(z)**2
# doesn't work...why?
np.dot(z, z)
# works
np.dot(z, z.T)

# identity matrix instantiation ...
x = np.identity(10)

# additional properties of matrices
x = np.arange(100).reshape(10,10)
y = np.arange(100, 200).reshape(10,10)
z = np.arange(200, 300).reshape(10, 10)

# 1
(x + y == y + x).sum() == x.ravel().shape[0]

# 2
(x + (y + z) == (x + y) + z).sum() == y.ravel().shape[0]

# 3
(x.dot(y.dot(z)) == np.matmul((np.matmul(x, y)), z)).sum() == z.ravel().shape[0]

# 4
x = x.reshape(20, 5)
y = y.reshape(5, 20)
z = z.reshape(5, 20)

(np.matmul(x, (y + z)) == x.dot(y) + x.dot(z)).sum() == x.shape[0]**2

# 5
x = x.reshape(50,2)
y = y.reshape(2, 50)
z = z.reshape(2, 50)

((y + z).dot(x) == y.dot(x) + z.dot(x)).sum() == x.shape[1]**2

# 6
x = x.reshape(20, 5)
a = 5
b = 10

((a + b)*x == (a*x + b*x)).sum() == x.shape[0] * x.shape[1]

# 7
x = x.reshape(20, 5)
y = y.reshape(20, 5)
a = 5

(a*(x + y) == (a*x + a*y)).sum() == y.flatten().shape[0]

# 8
x = x.reshape(20, 5)
y = y.reshape(5, 20)
a = 5

(x.dot(a*y) == a*(np.matmul(x, y))).sum() == x.shape[0]**2

# Eigendecomposition
x = np.diag(np.arange(11))
eig_vals, eig_vecs = np.linalg.eig(x)