Eigenvalue problems (eigs, eigsh) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of scipy.sparse.linalg.eigs on a symmetric matrix

What is the output of the following code snippet? It computes eigenvalues and eigenvectors of a symmetric matrix using eigs.

SciPy

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

A = np.array([[2, 1], [1, 2]])
vals, vecs = eigs(A, k=1, which='LM')
print(np.round(vals.real, 3))

A [0.0]

B [3.0]

C [2.0]

D [1.0]

💡 Hint

The matrix is symmetric with eigenvalues 3 and 1. 'which=LM' returns the largest magnitude eigenvalue.

The matrix [[2,1],[1,2]] has eigenvalues 3 and 1. Using eigs with k=1 and which='LM' returns the largest eigenvalue, which is 3.

❓ data_output

intermediate

Number of eigenvalues returned by eigsh

Given a 5x5 sparse symmetric matrix, how many eigenvalues and eigenvectors will eigsh return when called with k=3?

SciPy

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

A = np.diag(np.arange(1,6))
vals, vecs = eigsh(A, k=3)
print(len(vals), vecs.shape)

A 3 (5, 3)

B 5 (5, 5)

C 3 (3, 3)

D 5 (3, 5)

💡 Hint

The parameter k controls how many eigenvalues and eigenvectors are returned.

eigsh returns k eigenvalues and eigenvectors. For a 5x5 matrix with k=3, vals has length 3 and vecs shape is (5,3).

🔧 Debug

advanced

Error raised by eigs with k too large

What error does the following code raise? It tries to compute 5 eigenvalues of a 4x4 matrix using eigs.

SciPy

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

A = np.eye(4)
vals, vecs = eigs(A, k=5)

A TypeError: invalid argument type

B No error, returns 5 eigenvalues

C RuntimeWarning: convergence not achieved

D ValueError: k must be smaller than N

💡 Hint

k must be less than the size of the matrix.

eigs requires k < N where N is matrix size. Here k=5 and N=4, so ValueError is raised.

🧠 Conceptual

advanced

Difference between eigs and eigsh

Which statement correctly describes the difference between eigs and eigsh in scipy?

A <code>eigsh</code> is for symmetric or Hermitian matrices and is more efficient; <code>eigs</code> works for general matrices.

B <code>eigs</code> only works for symmetric matrices; <code>eigsh</code> works for any matrix.

C <code>eigs</code> uses dense matrix methods; <code>eigsh</code> uses sparse matrix methods.

D <code>eigsh</code> returns all eigenvalues; <code>eigs</code> returns only the largest eigenvalue.

💡 Hint

Think about matrix symmetry and algorithm optimization.

eigsh is optimized for symmetric or Hermitian matrices and is faster and more accurate for them. eigs works for any square matrix.

🚀 Application

expert

Identifying dominant eigenvalue with eigs

You have a large sparse matrix representing a network. You want to find the dominant eigenvalue (largest magnitude) and its eigenvector using eigs. Which code snippet correctly achieves this?

A vals, vecs = eigs(A, k=1, which='LR')

B vals, vecs = eigs(A, k=1, which='SM')

C vals, vecs = eigs(A, k=1, which='LM')

D vals, vecs = eigs(A, k=1, which='SR')

💡 Hint

'LM' means largest magnitude eigenvalue.

'LM' option in eigs returns eigenvalues with largest magnitude, which is the dominant eigenvalue.