Matrix and linear Algebra calculations in Python

In this article, we will learn Matrix and linear Algebra calculations in Python such as matrix multiplication, finding determinants, solving linear equations, etc.

A matrix object from the NumPy library can be used for this. When it comes to calculation, matrices are relatively comparable to the array objects. Linear Algebra is a huge topic, however, NumPy is an excellent library to start if you need to manipulate matrices and vectors.

Operations Covered

• Finding Transpose of a Matrix Using Numpy

• Finding Inverse of a Matrix Using Numpy

• Multiplying Matrix with a Vector

• Getting the Determinant of Matrix using numpy.linalg subpackage

• Finding Eigenvalues using numpy.linalg

• Solving equations using numpy.linalg

Finding Transpose of a Matrix

The numpy.matrix.T attribute returns the transpose of the given matrix ?

import numpy as np

# input matrix
inputMatrix = np.matrix([[6, 1, 5], [2, 0, 8], [1, 4, 3]])
print("Input Matrix:\n", inputMatrix)

# printing the transpose of an input matrix
print("Transpose of an input matrix:\n", inputMatrix.T)

Input Matrix:
 [[6 1 5]
 [2 0 8]
 [1 4 3]]
Transpose of an input matrix:
 [[6 2 1]
 [1 0 4]
 [5 8 3]]

Finding Inverse of a Matrix

The numpy.matrix.I attribute returns the inverse of the given matrix ?

import numpy as np

# input matrix 
inputMatrix = np.matrix([[6, 1, 5],[2, 0, 8],[1, 4, 3]])
print("Input Matrix:\n", inputMatrix)

# printing the inverse of an input matrix 
print("Inverse of an input matrix:\n", inputMatrix.I)

Input Matrix:
 [[6 1 5]
 [2 0 8]
 [1 4 3]]
Inverse of an input matrix:
 [[ 0.21333333 -0.11333333 -0.05333333]
 [-0.01333333 -0.08666667  0.25333333]
 [-0.05333333  0.15333333  0.01333333]]

Multiplying Matrix with a Vector

Matrix-vector multiplication uses the * operator with NumPy matrices ?

import numpy as np
 
# input matrix 
inputMatrix = np.matrix([[6, 1, 5],[2, 0, 8],[1, 4, 3]])
print("Input Matrix:\n", inputMatrix)

# creating a vector using numpy.matrix() function 
inputVector = np.matrix([[1],[3],[5]])

# printing the multiplication of the input matrix and vector 
print("Multiplication of input matrix and vector:\n", inputMatrix*inputVector)

Input Matrix:
 [[6 1 5]
 [2 0 8]
 [1 4 3]]
Multiplication of input matrix and vector:
 [[34]
 [42]
 [28]]

Getting the Determinant of Matrix

The numpy.linalg.det() function calculates the determinant of a square matrix ?

import numpy as np
 
# input matrix 
inputMatrix = np.matrix([[6, 1, 5],[2, 0, 8],[1, 4, 3]])
print("Input Matrix:\n", inputMatrix)

# getting the determinant of an input matrix 
outputDet = np.linalg.det(inputMatrix)

# printing the determinant of an input matrix 
print("Determinant of an input matrix:\n", outputDet)

Input Matrix:
 [[6 1 5]
 [2 0 8]
 [1 4 3]]
Determinant of an input matrix:
 -149.99999999999997

Finding Eigenvalues

The numpy.linalg.eigvals() function calculates the eigenvalues of a specified square matrix ?

import numpy as np
 
# input matrix 
inputMatrix = np.matrix([[6, 1, 5],[2, 0, 8],[1, 4, 3]])
print("Input Matrix:\n", inputMatrix)
 
# getting Eigenvalues of an input matrix 
eigenValues = np.linalg.eigvals(inputMatrix)
 
# printing Eigenvalues of an input matrix 
print("Eigenvalues of an input matrix:\n", eigenValues)

Input Matrix:
 [[6 1 5]
 [2 0 8]
 [1 4 3]]
Eigenvalues of an input matrix:
 [ 9.55480959  3.69447805 -4.24928765]

Solving Linear Equations

We can solve linear equation systems like A × X = B, where A is the matrix and B is the vector, using numpy.linalg.solve() ?

import numpy as np
 
# input matrix 
inputMatrix = np.matrix([[6, 1, 5],[2, 0, 8],[1, 4, 3]])
print("Input Matrix:\n", inputMatrix)
 
# creating a vector using np.matrix() function 
inputVector = np.matrix([[1],[3],[5]])
 
# getting the value of x in equation: inputMatrix * x = inputVector
x_value = np.linalg.solve(inputMatrix, inputVector)
 
print("x value:\n", x_value)
 
# verification: multiplying input matrix with x values 
print("Verification (A × X):\n", inputMatrix * x_value)

Input Matrix:
 [[6 1 5]
 [2 0 8]
 [1 4 3]]
x value:
 [[-0.39333333]
 [ 0.99333333]
 [ 0.47333333]]
Verification (A × X):
 [[1.]
 [3.]
 [5.]]

Conclusion

NumPy provides comprehensive tools for matrix and linear algebra operations including transpose, inverse, determinant, eigenvalues, and solving linear equation systems. These functions form the foundation for more advanced numerical computing and data science applications.