1.3 linear algebra

1.3.1 Scalar

# Scalars are represented by tensors with only one element 
from mxnet import np, npx
npx.set_np()
x = np.array(3.0)
y = np.array(2.0)
x + y, x * y, x / y, x**y

(array(5.), array(6.), array(1.5), array(9.))

1.3.2 vector

x = np.arange(4)
x

array([0., 1., 2., 3.])

len(x)

4

x.shape

(4,)

x = np.array([[0],[1],[2],[3]])
x.shape

(4, 1)

1.3.3 matrix

A = np.arange(20).reshape(5,4)
A

array([[ 0.,  1.,  2.,  3.],
       [ 4.,  5.,  6.,  7.],
       [ 8.,  9., 10., 11.],
       [12., 13., 14., 15.],
       [16., 17., 18., 19.]])

A.T

array([[ 0.,  4.,  8., 12., 16.],
       [ 1.,  5.,  9., 13., 17.],
       [ 2.,  6., 10., 14., 18.],
       [ 3.,  7., 11., 15., 19.]])

1.3.4 tensor

X = np.arange(24).reshape(2,3,4)
X

array([[[ 0.,  1.,  2.,  3.],
        [ 4.,  5.,  6.,  7.],
        [ 8.,  9., 10., 11.]],

       [[12., 13., 14., 15.],
        [16., 17., 18., 19.],
        [20., 21., 22., 23.]]])

1.3.5 Basic properties of tensor algorithm

A = np.arange(20).reshape(5,4)
B = A.copy()
A, A + B

(array([[ 0.,  1.,  2.,  3.],
        [ 4.,  5.,  6.,  7.],
        [ 8.,  9., 10., 11.],
        [12., 13., 14., 15.],
        [16., 17., 18., 19.]]),
 array([[ 0.,  2.,  4.,  6.],
        [ 8., 10., 12., 14.],
        [16., 18., 20., 22.],
        [24., 26., 28., 30.],
        [32., 34., 36., 38.]]))

# The multiplication of two matrices by elements is called dahama product 
A * B, A**2

(array([[  0.,   1.,   4.,   9.],
        [ 16.,  25.,  36.,  49.],
        [ 64.,  81., 100., 121.],
        [144., 169., 196., 225.],
        [256., 289., 324., 361.]]),
 array([[  0.,   1.,   4.,   9.],
        [ 16.,  25.,  36.,  49.],
        [ 64.,  81., 100., 121.],
        [144., 169., 196., 225.],
        [256., 289., 324., 361.]]))

# Multiplying the tensor by or adding a scalar does not change the shape of the tensor , Each element of the tensor will be multiplied by or added to the scalar 
a = 2
X = np.arange(24).reshape(2,3,4)
a + X, (a * X).shape

(array([[[ 2.,  3.,  4.,  5.],
         [ 6.,  7.,  8.,  9.],
         [10., 11., 12., 13.]],
 
        [[14., 15., 16., 17.],
         [18., 19., 20., 21.],
         [22., 23., 24., 25.]]]),
 (2, 3, 4))

1.3.6 Summary

x = np.arange(4)
x, x.sum()

(array([0., 1., 2., 3.]), array(6.))

A, A.shape, A.sum()

(array([[ 0.,  1.,  2.,  3.],
        [ 4.,  5.,  6.,  7.],
        [ 8.,  9., 10., 11.],
        [12., 13., 14., 15.],
        [16., 17., 18., 19.]]),
 (5, 4),
 array(190.))

A_sum_axis0 = A.sum(axis = 0)
A_sum_axis0, A_sum_axis0.shape

(array([40., 45., 50., 55.]), (4,))

A_sum_axis1 = A.sum(axis = 1)
A_sum_axis1, A_sum_axis1.shape

(array([ 6., 22., 38., 54., 70.]), (5,))

A.sum(axis = [0,1]) # Sum along the row and column 

# averaging 
A.mean(), A.sum() / A.size

(array(9.5), array(9.5))

A.mean(axis = 0), A.sum(axis = 0) / A.shape[0]

(array([ 8.,  9., 10., 11.]), array([ 8.,  9., 10., 11.]))

# Non summary summation , Keep the number of axes unchanged when calling the function to calculate the sum or average 
sum_A = A.sum(axis = 1, keepdims = True)
sum_A

array([[ 6.],
       [22.],
       [38.],
       [54.],
       [70.]])

A / sum_A

array([[0.        , 0.16666667, 0.33333334, 0.5       ],
       [0.18181819, 0.22727273, 0.27272728, 0.3181818 ],
       [0.21052632, 0.23684211, 0.2631579 , 0.28947368],
       [0.22222222, 0.24074075, 0.25925925, 0.2777778 ],
       [0.22857143, 0.24285714, 0.25714287, 0.27142859]])

# Calculate along an axis A The cumulative sum of elements 
A.cumsum(axis = 0)

array([[ 0.,  1.,  2.,  3.],
       [ 4.,  6.,  8., 10.],
       [12., 15., 18., 21.],
       [24., 28., 32., 36.],
       [40., 45., 50., 55.]])

1.3.7 Dot product

y = np.ones(4)
x, y, np.dot(x, y), np.sum(x * y)

(array([0., 1., 2., 3.]), array([1., 1., 1., 1.]), array(6.), array(6.))

1.3.8 matrix - Vector product

A.shape, x.shape, np.dot(A, x)

((5, 4), (4,), array([ 14.,  38.,  62.,  86., 110.]))

1.3.9 matrix - Matrix multiplication

B = np.ones(shape = (4,3))
np.dot(A, B)

array([[ 6.,  6.,  6.],
       [22., 22., 22.],
       [38., 38., 38.],
       [54., 54., 54.],
       [70., 70., 70.]])

1.3.10 norm

u = np.array([3,-4])
np.linalg.norm(u)

array(5.)

np.abs(u).sum()

array(7.)

np.linalg.norm(np.ones((4,9)))

array(6.)