Matrix and linear algebra calculations in Python

In this article, we will learn how to use Python to perform matrix and linear algebra calculations, such as matrix multiplication, finding determinants, solving linear equations, etc.

You can use a matrix object from the NumPy library to achieve this. When doing calculations, matrices are relatively comparable to array objects.

Linear algebra is a vast subject and beyond the scope of this article.

However, if you need to manipulate matrices and vectors, NumPy is a good starting point.

usage instructions

• Find the transpose of a matrix using Numpy

• Find the inverse of a matrix using Numpy

• Matrix and vector multiplication

• Use the numpy.linalg subpackage to obtain the determinant of the matrix

• Use numpy.linalg to find eigenvalues

• Use numpy.linalg to solve equations

Method 1: Find the transpose of a matrix using Numpy

numpy.matrix.T Properties − Returns the transpose of the given matrix.

The Chinese translation of

Example

is:

Example

The following program uses the numpy.matrix.T property to return the transpose of the matrix −

# importing NumPy module
import numpy as np

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

# printing the transpose of an input matrix
# by applying the .T attribute of the NumPy matrix of the numpy Module
print("Transpose of an input matrix\n", inputMatrix.T)

Output

When executed, the above program will generate the following output -

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]]

Method 2: Find the inverse of a matrix using Numpy

numpy.matrix.I Properties - Returns the inverse of the given matrix.

The Chinese translation of

Example

is:

Example

The following program uses the numpy.matrix.I property to return the inverse of the matrix −

# importing NumPy module 
import numpy as np

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

# printing the inverse of an input matrix 
# by applying the .I attribute of the NumPy matrix of the numpy Module
print("Inverse of an input matrix:\n", inputMatrix.I)

Output

When executed, the above program will generate the following output -

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]]

Method 3: Multiplying matrices and vectors

The Chinese translation of

Example

is:

Example

The following program uses the * operator to return the product of the input matrix and vector -

# importing numpy module 
import numpy as np
 
# input matrix 
inputMatrix = np.matrix([[6, 1, 5],[2, 0, 8],[1, 4, 3]])
# printing the input matrix
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)

Output

When executed, the above program will generate the following output -

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

Method 4: Use the numpy.linalg subpackage to obtain the determinant of the matrix

numpy.linalg.det() Function − Calculate the determinant of a square matrix.

The Chinese translation of

Example

is:

Example

The following program uses the numpy.linalg.det() function to return the determinant of the matrix −

# importing numpy module 
import numpy as np
 
# input matrix 
inputMatrix = np.matrix([[6, 1, 5],[2, 0, 8],[1, 4, 3]])
# printing the input matrix
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)

Output

When executed, the above program will generate the following output -

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

The fifth way to find eigenvalues ​​using numpy.linalg

numpy.linalg.eigvals() function − Calculate the eigenvalues ​​and right eigenvectors of the specified square matrix/matrix.

The Chinese translation of

Example

is:

Example

The following program returns the Eigenvalues ​​of an input matrix using the numpy.linalg.eigvals() function −

# importing NumPy module 
import numpy as np
 
# input matrix 
inputMatrix = np.matrix([[6, 1, 5],[2, 0, 8],[1, 4, 3]])
# printing the input matrix
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)

Output

When executed, the above program will generate the following output -

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

Method 6: Use numpy.linalg to solve equations

We can solve a problem similar to finding the value of X for A*X = B,

Where A is a matrix and B is a vector.

The Chinese translation of

Example

is:

Example

The following is a program that uses the solve() function to return the x value-

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

Output

When executed, the above program will generate the following output -

Input Matrix:
 [[6 1 5]
 [2 0 8]
 [1 4 3]]
x value:
 [[-0.39333333]
 [ 0.99333333]
 [ 0.47333333]]
Multiplication of input matrix with x values:
 [[1.]
 [3.]
 [5.]]

in conclusion

In this article, we learned how to perform matrix and linear algebra operations using the NumPy module in Python. We learned how to calculate the transpose, inverse, and determinant of a matrix. We also learned how to do some calculations in linear algebra, such as solving equations and determining eigenvalues.

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