In this post, you will learn about how to calculate **Eigenvalues **and **Eigenvectors **using **Python code examples. **Before getting ahead and learning the code examples, you may want to check out this post on **when & why to use Eigenvalues and Eigenvectors. **As a machine learning Engineer / Data Scientist, you must get a good understanding of Eigenvalues / Eigenvectors concepts as it proves to be very useful in feature extraction techniques such as principal components analysis. Python **Numpy** package is used for illustration purpose. The following topics are covered in this post:

- Creating Eigenvectors / Eigenvalues using Numpy Linalg module
- Re-creating original transformation matrix from eigenvalues & eigenvectors

Table of Contents

## Creating Eigenvectors / Eigenvalues using Numpy

In this section, you will learn about how to create Eigenvalues and Eigenvectors for a given square matrix (transformation matrix) using Python Numpy library. Here are the steps:

- Create a sample Numpy array representing a set of dummy independent variables / features
- Scale the features
- Calculate the n x n covariance matrix. Note that the
**transpose**of the matrix is taken.**One can use np.cov(students_scaled, rowvar=False) instead to represent that columns represent the variables**. - Calculate the eigenvalues and eigenvectors using Numpy
**linalg.eig**method. This method is designed to operate on both symmetric and non-symmetric square matrices. There is another method such as**linalg.eigh**which is used to decompose Hermitian matrices which is nothing but a complex square matrix that is equal to its own conjugate transpose. The**linalg.eigh**method is considered to be numerically more stable approach to working with symmetric matrices such as the covariance matrix.

```
import numpy as np
from sklearn.preprocessing import StandardScaler
from numpy.linalg import eig
#
# Percentage of marks and no. of hours studied
#
students = np.array([[85.4, 5],
[82.3, 6],
[97, 7],
[96.5, 6.5]])
#
# Scale the features
#
sc = StandardScaler()
students_scaled = sc.fit_transform(students)
#
# Calculate covariance matrix; One can also use the following
# code: np.cov(students_scaled, rowvar=False)
#
cov_matrix = np.cov(students_scaled.T)
#
# Calculate Eigenvalues and Eigenmatrix
#
eigenvalues, eigenvectors = eig(cov_matrix)
```

Here is how the output of above looks like:

Let’s confirm whether the above is correct by calculating LHS and RHS of the following and making sure that LHS = RHS. A represents the transformation matrix (cob_matrix in above example), x represents eigenvectors and \(\lambda\) represents eigenvalues

\( Ax = \lambda x\)

Here is the code comparing LHS to RHS

```
#
# LHS
#
cov_matrix.dot(eigenvectors[:, 0])
#
# RHS
#
eigenvalues[0]*eigenvectors[:, 0]
```

From the output represented in the picture below, it does confirm that above calculation done by **Numpy** **linalg.eig** method is correct.

## Conclusion

Here is what you learned in this post:

- One will require to scale the data before calculating its Eigenvalues and Eigenvectors
- One will need to have the transformation matrix as square matrix N x N representing N dimensions in order to calculate N eigenvalues and Eigenvectors
- Numpy linear algebra module
**linalg**can be used along with**eig**to determine Eigenvalues and Eigenvectors.

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