Eigenvalue problems (eigs, eigsh) in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is an eigenvalue in the context of matrices?

An eigenvalue is a special number associated with a matrix that shows how a vector changes when the matrix is applied to it. Specifically, if you multiply the matrix by a vector and the result is the same vector scaled by this number, that number is the eigenvalue.

beginner

What does the function scipy.sparse.linalg.eigs do?

The function eigs finds a few eigenvalues and eigenvectors of a square matrix, especially when the matrix is large and sparse. It works for general (not necessarily symmetric) matrices.

intermediate

When should you use scipy.sparse.linalg.eigsh instead of eigs?

eigsh is designed for symmetric or Hermitian matrices. It is faster and more accurate for these types of matrices compared to eigs.

beginner

What does the parameter k specify in eigs and eigsh?

The parameter k tells the function how many eigenvalues and eigenvectors to find. For example, k=3 means find 3 eigenvalues and their vectors.

beginner

Why might you want to find only a few eigenvalues instead of all of them?

For very large matrices, finding all eigenvalues is slow and uses a lot of memory. Often, only the largest or smallest eigenvalues are important for understanding the system, so finding just a few saves time and resources.

Which function is best for finding eigenvalues of a symmetric matrix?

A scipy.sparse.linalg.eigs

B scipy.sparse.linalg.eigsh

C numpy.linalg.eig

D scipy.linalg.inv

eigsh is optimized for symmetric or Hermitian matrices, making it the best choice here.

What does the k parameter control in eigs and eigsh?

A Number of eigenvalues to find

B Matrix size

C Tolerance level

D Maximum iterations

k sets how many eigenvalues and eigenvectors the function returns.

If you have a large, non-symmetric matrix, which function should you use?

A scipy.sparse.linalg.eigsh

B numpy.linalg.eigh

C scipy.linalg.det

D scipy.sparse.linalg.eigs

eigs works for general matrices, including non-symmetric ones.

Why is it often unnecessary to compute all eigenvalues for large matrices?

A Because only a few eigenvalues are usually important

B Because eigenvalues are always the same

C Because matrices have no eigenvalues

D Because computing eigenvalues is always fast

Usually, only the largest or smallest eigenvalues give useful information, so computing all is wasteful.

What type of matrix is required for eigsh to work correctly?

A Non-square

B Diagonal

C Symmetric or Hermitian

D Sparse only

eigsh requires the matrix to be symmetric or Hermitian for accurate results.

Explain the difference between eigs and eigsh in SciPy and when to use each.

Think about matrix symmetry and performance.

Describe why finding only a few eigenvalues is useful in data science or engineering problems.

Consider practical reasons and examples.