Contents

• NumPy
  • 1 Creating arrays
  • 2 Operations
  • 3 Related functions
  • 4 Practice

Numpy

import numpy as np

1 Creating arrays

• Special matrices
  • All-1 matrix, all-0 matrix, identity matrix

# Declare a 3x2 matrix of 1s
# Note that the Input is a tuple!
one_matrix = np.ones((3,2)) 
print(one_matrix)

[[1. 1.]
 [1. 1.]
 [1. 1.]]

# Declare a 3x2 matrix of 0s
zero_matrix = np.zeros((3,2))
print(zero_matrix)

[[0. 0.]
 [0. 0.]
 [0. 0.]]

# Declare a 3x3 identity matrix
eye_matrix = np.eye(3)
print(eye_matrix)

[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]

• General matrices
  • Enter elements directly and change the data type.

# Declare a matrix using a list
data_int = np.array([[1,2,3],[4,5,6],[7,8,9]])
print(data_int)

[[1 2 3]
 [4 5 6]
 [7 8 9]]

# Print the Matrix shape
print(data_int.shape)

(3, 3)

# Print the element dtype
print(data_int.dtype)

int64

# Change the dtype
data_float = data_int.astype(np.float64)
print(data_float)
print(data_float.dtype)

[[1. 2. 3.]
 [4. 5. 6.]
 [7. 8. 9.]]
float64

2 Matrix operations

• Arithmetic - Element-wise operations

# Element-wise multiplication
print(data_int * data_int)

[[ 1  4  9]
 [16 25 36]
 [49 64 81]]

# Add a constant
print(data_int + 3)

[[ 4  5  6]
 [ 7  8  9]
 [10 11 12]]

# Divide by a constant
print(data_int / 3)

[[0.33333333 0.66666667 1.        ]
 [1.33333333 1.66666667 2.        ]
 [2.33333333 2.66666667 3.        ]]

• Matrix transpose

x = np.array([[1,2,3],[4,5,6]])
y = np.array([[1,0,-1],[1,1,0]])
print(x)
print(y)
print(y.T)

[[1 2 3]
 [4 5 6]]
[[ 1  0 -1]
 [ 1  1  0]]
[[ 1  1]
 [ 0  1]
 [-1  0]]

• Matrix multiplication - Dot product

print(np.dot(x,y.T))

[[-2  3]
 [-2  9]]

• Select and modify matrix elements

# Create an array of consecutive integers
data_range = np.arange(10)
print(data_range)

[0 1 2 3 4 5 6 7 8 9]

# Select using an index
print(data_range[0]) 

0

# slicing
print(data_range[1:4])

[1 2 3]

# Modify using slicing
data_range[1:4] = 10
print(data_range)

[ 0 10 10 10  4  5  6  7  8  9]

# Note that slicing returns an address (pointer)!
tmp = data_range[1:4]
tmp[1] = 20
print(data_range)

[ 0 10 20 10  4  5  6  7  8  9]

# The same applies to 2 or more dimensions
print(data_int[2])
print(data_int[:2])
print(data_int[2][0])
print(data_int[2,0])
print(data_int[:2,1:])
data_int[:2,1:] = 0
print(data_int)

[7 8 9]
[[1 2 3]
 [4 5 6]]
7
7
[[2 3]
 [5 6]]
[[1 0 0]
 [4 0 0]
 [7 8 9]]

3 Related functions

• Compute the inverse matrix

import numpy.linalg as lin
mat = np.array([[2,2,0],[-2,1,1],[3,0,1]])
print(lin.inv(mat))

[[ 0.08333333 -0.16666667  0.16666667]
 [ 0.41666667  0.16666667 -0.16666667]
 [-0.25        0.5         0.5       ]]

• Compute the determinant

print(lin.det(mat))

12.0

• Solve a matrix equation $$Ax=B$$ $$\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix}b_1 \\ b_2 \\ b_3 \end{bmatrix}$$

matA = np.array([[3,2,1],[1,-1,3],[5,4,-2]])
matB = np.array([7,3,1])
print(lin.solve(matA, matB))

[-3.  6.  4.]

4 Practice (Assignment 3)

• Solve a matrix equation
  • Solve the following matrix equation using the pseudoinverse.
  • Assume that $A^{T}A$ is invertible. $$Ax=B$$ $$A = \begin{bmatrix}0 & 1 \\ 1 & 1 \\ 2 & 1 \\ 3 & 1 \end{bmatrix}$$ $$B = \begin{bmatrix}-1 \\ 0.2 \\ 0.9 \\ 2.1 \end{bmatrix}$$