Eigenvalue problems (eigs, eigsh) in SciPy

Introduction

Eigenvalue problems help us find special numbers and vectors that describe important properties of a matrix. This is useful to understand systems like vibrations, stability, or data patterns.

Finding the main directions of data in Principal Component Analysis (PCA).

Analyzing vibrations in mechanical structures to find natural frequencies.

Solving large systems where only a few eigenvalues and eigenvectors are needed.

Studying stability of systems by checking eigenvalues of matrices.

Reducing data dimensions by focusing on important eigenvectors.

Syntax

SciPy

from scipy.sparse.linalg import eigs, eigsh

# eigs for general square matrices
values, vectors = eigs(A, k=number_of_eigenvalues)

# eigsh for symmetric or Hermitian matrices
values, vectors = eigsh(A, k=number_of_eigenvalues)

eigs works for any square matrix but is slower for symmetric ones.

eigsh is faster and more accurate for symmetric or Hermitian matrices.

Examples

Finds the largest eigenvalue of a 2x2 matrix using eigs.

SciPy

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

A = np.array([[1, 2], [3, 4]])
values, vectors = eigs(A, k=1)
print(values)

Finds the largest eigenvalue of a symmetric matrix using eigsh.

SciPy

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

A = np.array([[2, 1], [1, 2]])
values, vectors = eigsh(A, k=1)
print(values)

Finds the two largest eigenvalues of a symmetric matrix.

SciPy

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

A = np.array([[2, 1], [1, 2]])
values, vectors = eigsh(A, k=2)
print(values)

Sample Program

This program shows how to use eigs for a general matrix and eigsh for a symmetric matrix. It prints the eigenvalues found.

SciPy

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

# Define a general matrix
A = np.array([[4, 2], [1, 3]])

# Use eigs to find the largest eigenvalue and eigenvector
values_eigs, vectors_eigs = eigs(A, k=1)

# Define a symmetric matrix
B = np.array([[2, 1], [1, 2]])

# Use eigsh to find the two largest eigenvalues and eigenvectors
values_eigsh, vectors_eigsh = eigsh(B, k=2)

print("Largest eigenvalue using eigs:", values_eigs)
print("Eigenvalues using eigsh:", values_eigsh)

Important Notes

Always use eigsh for symmetric matrices for better speed and accuracy.

The k parameter controls how many eigenvalues you want to find.

Eigenvalues can be complex numbers when using eigs on non-symmetric matrices.

Summary

Eigenvalue problems find special numbers (eigenvalues) and vectors (eigenvectors) that reveal matrix properties.

eigs works for any square matrix; eigsh is optimized for symmetric matrices.

Use these functions to analyze data, systems, or reduce dimensions efficiently.

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