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

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

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

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

Hint

Use two print statements to show eigenvalues and eigenvectors.

Practice

1. What is the main difference between scipy.sparse.linalg.eigs and scipy.sparse.linalg.eigsh?

easy

A. eigs returns eigenvectors only, while eigsh returns eigenvalues only.

B. eigs works for any square matrix, while eigsh is optimized for symmetric or Hermitian matrices.

C. eigsh works for any square matrix, while eigs only works for diagonal matrices.

D. eigsh is used for non-square matrices, while eigs is for square matrices.

Solution

1. Step 1: Understand the function purposes

   eigs is designed to find eigenvalues and eigenvectors of any square matrix, including non-symmetric ones. eigsh is a specialized version optimized for symmetric or Hermitian matrices, which are common in many applications.

2. Step 2: Compare matrix types each function supports

   eigsh takes advantage of symmetry to be faster and more accurate, but it requires the matrix to be symmetric. eigs has no such restriction but may be slower.

3. Final Answer:

   eigs works for any square matrix, while eigsh is optimized for symmetric or Hermitian matrices. -> Option B

4. Quick Check:

   Function specialization = C [OK]

Hint: Remember: eigsh = symmetric only, eigs = any square matrix [OK]

Common Mistakes:

• Thinking eigsh works for any matrix
• Confusing eigs and eigsh outputs
• Assuming eigsh works for non-square matrices

2. Which of the following is the correct way to import and use eigsh from scipy.sparse.linalg to compute 3 eigenvalues of a symmetric matrix A?

easy

A. from scipy.sparse.linalg import eigsh

vals, vecs = eigsh(A, k=3)

B. import scipy.linalg as la

vals, vecs = la.eigsh(A, 3)

C. from scipy.linalg import eigsh

vals, vecs = eigsh(A, 3)

D. from scipy.sparse.linalg import eigs

vals, vecs = eigs(A, k=3)

Solution

1. Step 1: Check the correct import statement

   eigsh is in scipy.sparse.linalg, so the import must be from there, not scipy.linalg.

2. Step 2: Verify function call syntax

   The function call requires the matrix A and the number of eigenvalues k=3. from scipy.sparse.linalg import eigsh vals, vecs = eigsh(A, k=3) uses correct syntax and import.

3. Final Answer:

   from scipy.sparse.linalg import eigsh vals, vecs = eigsh(A, k=3) -> Option A

4. Quick Check:

   Correct import and call = D [OK]

Hint: Import eigsh from scipy.sparse.linalg and use k=number [OK]

Common Mistakes:

• Importing eigsh from scipy.linalg instead of scipy.sparse.linalg
• Using eigs instead of eigsh for symmetric matrices
• Passing number without keyword k

3. Given the code below, what will be the output of print(vals)?

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

A = np.array([[2, 1], [1, 2]])
vals, vecs = eigsh(A, k=1, which='LM')
print(np.round(vals, 2))

medium

A. [2.00]

B. [1.00]

C. [3.00]

D. [0.00]

Solution

1. Step 1: Understand the matrix and eigenvalues

   Matrix A is symmetric with values [[2,1],[1,2]]. Its eigenvalues are 3 and 1.

2. Step 2: Check the function call parameters

   eigsh is called with k=1 and which='LM' meaning largest magnitude eigenvalue. So it returns the largest eigenvalue, which is 3.

3. Final Answer:

   [3.00] -> Option C

4. Quick Check:

   Largest eigenvalue = 3.00 [OK]

Hint: which='LM' returns largest eigenvalue [OK]

Common Mistakes:

• Confusing largest eigenvalue with smallest
• Not rounding output
• Using eigs instead of eigsh for symmetric matrix

4. The following code raises an error. What is the most likely cause?

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

A = np.array([[1, 2], [3, 4]])
vals, vecs = eigsh(A, k=1)

medium

A. The import statement is incorrect.

B. The value of k is too large for the matrix size.

C. eigsh requires the matrix to be sparse, but A is dense.

D. Matrix A is not symmetric, so eigsh cannot be used.

Solution

1. Step 1: Check matrix properties

   Matrix A = [[1,2],[3,4]] is not symmetric because A[0,1] != A[1,0].

2. Step 2: Understand eigsh requirements

   eigsh requires the matrix to be symmetric or Hermitian. Using it on a non-symmetric matrix causes an error.

3. Final Answer:

   Matrix A is not symmetric, so eigsh cannot be used. -> Option D

4. Quick Check:

   Symmetry required for eigsh = A [OK]

Hint: Check matrix symmetry before using eigsh [OK]

Common Mistakes:

• Assuming eigsh works on any matrix
• Thinking k=1 is too large for 2x2 matrix
• Believing eigsh only works on sparse matrices

5. You have a large symmetric matrix representing connections in a social network. You want to find the 5 smallest eigenvalues to analyze community structure. Which approach is best?

hard

A. Use eigsh with k=5 and which='SM' to get the smallest eigenvalues efficiently.

B. Use eigs with k=5 and which='LM' to get the largest eigenvalues.

C. Convert the matrix to dense and use numpy.linalg.eig to get all eigenvalues.

D. Use eigsh with k=5 and which='LM' to get the largest eigenvalues.

Solution

1. Step 1: Identify matrix type and goal

   The matrix is large and symmetric, and we want the 5 smallest eigenvalues to study community structure.

2. Step 2: Choose appropriate function and parameters

   eigsh is optimized for symmetric matrices. Using k=5 and which='SM' returns the smallest magnitude eigenvalues efficiently without computing all eigenvalues.

3. Step 3: Evaluate other options

   eigs is less efficient for symmetric matrices. Converting to dense is costly for large matrices. Getting largest eigenvalues is not the goal.

4. Final Answer:

   Use eigsh with k=5 and which='SM' to get the smallest eigenvalues efficiently. -> Option A

5. Quick Check:

   Symmetric + smallest eigenvalues = eigsh + which='SM' [OK]

Hint: For smallest eigenvalues of symmetric matrix, use eigsh with which='SM' [OK]

Common Mistakes:

• Using eigs instead of eigsh for symmetric matrix
• Requesting largest eigenvalues instead of smallest
• Converting large sparse matrix to dense unnecessarily