Exploring Linear Algebra with Python

25 September 2017 — Written by Mubaris NK

Scalars, Vectors and Matrices

Scalars, Vectors and Matrices are the basic objects in Linear Algebra

Scalar: A scalar is just a number.
Vector: A vector is an array of numbers and it's order is important.
Matrix: A Matrix is a 2D array of numbers.

NumPy is a python package that can be used for Linear Algebra calculations. We can use NumPy to create Vectors and Matrices in Python. You don't need any special packages to create Scalar, since it's just a number.

# Import NumPy
import numpy as np

# Scalar
s = 32

# Vectors
a = np.array([1, 2, 3])
b = np.array([3, 4, 5])

# Matrices
A = np.array([
    [3, 5, 7],
    [4, 6, 8]
])
B = np.array([
    [4, 7],
    [5, 8],
    [6, 9]
])

# Size of the Matrices
print(A.shape) # (2, 3)
print(B.shape) # (3, 2)

Operations

Many operations can applied on linear algebra objects.

Addition

To add 2 vectors they should have same number of elements. And to add 2 matrices they sould have same size(shape).

# Addition of vectors
c = a + b
print(c) # [4, 6, 8]

# Addition of matrices
X = np.array([
    [1, 2, 3],
    [4, 5, 6]
])
Y = np.array([
    [7, 8, 9],
    [1, 2, 3]
])
Z = X + Y
print(Z) # [[8, 10, 12], [5, 7, 9]]

Subtraction

To subtract 2 vectors they should have same number of elements and to subtract 2 matrices they should have size(shape).

# Subtraction of vectors
d = a - b
print(d) # [-2, -2, -2]

# Subtraction of matrices
W = X - Y
print(W) # [[-6, -6, -6], [3, 3, 3]]

Transpose of a Matrix

Transpose of a Matrix is an operator which flips a matrix over its main diagonal like a mirror image. This can be done by calling numpy.transpose function or T method in numpy.

print(X.T) # [[1, 4], [2, 5], [3, 6]]
print(np.transpose(Y)) # [[7, 1], [8, 2], [9, 3]]

# Finding shape of transpose
print(X.shape) # (2, 3)
print(X.T.shape) # (3, 2)

print(Y.shape) # (2, 3)
print(np.transpose(Y).shape) # (3, 2)

Multiplication

NumPy uses numpy.dot function for multiplication os both vectors and matrices. Matrix multiplication is not commutative.

# Vectors
e = np.dot(a, b)
f = np.dot(b, a)
print(e) # 26
print(f) # 26

# Matrices
C = np.dot(A, B)
D = np.dot(B, A)
print(C) # [[79, 124], [94, 148]]
print(D) # [[40, 62, 84], [47, 73, 99], [54, 84, 114]]

# Size of C and D
print(C.shape) # (2, 2)
print(D.shape) # (3, 3)

# Vectors and Matrices
g = np.dot(A, b) # [64, 76]

Inverse

Inverse operation only applies to sqaure matrices(matrix with same number of columns and rows). To compute inverse using numpy we need to use numpy.linalg.inv function.

P = np.array([
    [1, 3, 3],
    [1, 4, 3],
    [1, 3, 4]
])
Pinv = np.linalg.inv(P)
print(Pinv) # [[7, -3, -3], [-1, 1, 0], [-1, 0, 1]]

Types of Matrices

Diagonal Matrix

A diagonal matrix is a square matrix which has zeros everywhere other than main diagonal.

Scalar Matrix

A scalar matrix is a diagonal matrix whose diagonal values are same.

Identity Matrix

An Identity Matrix is a scalar matrix whose diagonal values are 1.

Zero Matrix

A zero matrix is matrix which has zeros everywhere.

Implementation

# Identity Matrix of size (3, 3)
I3 = np.identity(3)
# Identity Matrix of size (2, 2)
I2 = np.identity(2)

# Zero Matrix
Q = np.zeros((3, 2)) # Size (3, 2)
R = np.zeros((5, 6)) # Size (5, 6)

Determinant

Determinant of matrix is a special value that can be calculated from a square matrix.

A = np.array([
    [1, 3, 3],
    [4, 5, 6],
    [7, 8, 9]
])
B = np.array([
    [34, 54],
    [67, 87]
])

adet = np.linalg.det(A)
bdet = np.linalg.det(B)

print(adet) # 6.0
print(bdet) # -660.0