Eigenvalue problems (eigs, eigsh) in SciPy - Mini Project: Build & Apply

Finding Eigenvalues and Eigenvectors with SciPy

📖 Scenario: Imagine you are analyzing a simple mechanical system where you want to find its natural vibration modes. These modes are found by calculating eigenvalues and eigenvectors of a matrix representing the system.

🎯 Goal: You will create a matrix representing the system, configure how many eigenvalues to find, compute the eigenvalues and eigenvectors using SciPy's eigs function, and finally display the results.

📋 What You'll Learn

Create a 3x3 matrix called A with specific values

Create a variable num_eigenvalues to specify how many eigenvalues to compute

Use scipy.sparse.linalg.eigs to compute eigenvalues and eigenvectors

Print the eigenvalues and eigenvectors

💡 Why This Matters

🌍 Real World

Eigenvalue problems are used in physics, engineering, and data science to understand system behaviors like vibrations, stability, and principal components.

💼 Career

Knowing how to compute eigenvalues and eigenvectors is important for roles in data analysis, machine learning, and engineering simulations.

1

Create the matrix A

Create a 3x3 NumPy array called A with these exact values: [[6, 2, 1], [2, 3, 1], [1, 1, 1]]

SciPy

import numpy as np
from scipy.sparse.linalg import eigs

# Create the matrix A with the given values
# Your code here

Need a hint?

Use np.array and pass the list of lists exactly as shown.

2

Set the number of eigenvalues to find

Create a variable called num_eigenvalues and set it to 2 to find two eigenvalues

SciPy

import numpy as np
from scipy.sparse.linalg import eigs

A = np.array([[6, 2, 1], [2, 3, 1], [1, 1, 1]])

# Set the number of eigenvalues to find
# Your code here

Need a hint?

Just assign the number 2 to the variable num_eigenvalues.

3

Compute eigenvalues and eigenvectors using eigs

Use eigs from scipy.sparse.linalg to compute num_eigenvalues eigenvalues and eigenvectors of A. Store the eigenvalues in eigenvalues and eigenvectors in eigenvectors

SciPy

import numpy as np
from scipy.sparse.linalg import eigs

A = np.array([[6, 2, 1], [2, 3, 1], [1, 1, 1]])
num_eigenvalues = 2

# Compute eigenvalues and eigenvectors
# Your code here

Need a hint?

Call eigs with A and k=num_eigenvalues, and unpack the result into eigenvalues and eigenvectors.

4

Print the eigenvalues and eigenvectors

Print the variables eigenvalues and eigenvectors to display the results

SciPy

import numpy as np
from scipy.sparse.linalg import eigs

A = np.array([[6, 2, 1], [2, 3, 1], [1, 1, 1]])
num_eigenvalues = 2

eigenvalues, eigenvectors = eigs(A, k=num_eigenvalues)

# Print the eigenvalues and eigenvectors
# Your code here

Need a hint?

Use two print statements to show eigenvalues and eigenvectors.