Eigenvalue problems (eigs, eigsh) in SciPy - Step-by-Step Execution

Concept Flow - Eigenvalue problems (eigs, eigsh)

Start: Define matrix A

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Choose eigs or eigsh

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Call eigs/eigsh with A

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Compute eigenvalues and eigenvectors

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Return eigenvalues, eigenvectors

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Use results for analysis or visualization

We start with a matrix, pick the right solver (eigs for general, eigsh for symmetric), compute eigenvalues and eigenvectors, then use them.

Execution Sample

SciPy

import numpy as np
from scipy.sparse.linalg import eigsh
A = np.array([[2, 1], [1, 2]])
vals, vecs = eigsh(A, k=1)
print(vals, vecs)

This code finds the largest eigenvalue and eigenvector of matrix A using eigsh.

Execution Table

Step	Action	Matrix A	k (number of eigenvalues)	Eigenvalues computed	Eigenvectors computed
1	Define matrix A	[[2 1] [1 2]]	N/A	N/A	N/A
2	Choose eigsh solver	[[2 1] [1 2]]	1	N/A	N/A
3	Call eigsh(A, k=1)	[[2 1] [1 2]]	1	N/A	N/A
4	Compute eigenvalues and eigenvectors	[[2 1] [1 2]]	1	[3.0]	[[0.70710678] [0.70710678]]
5	Return results	[[2 1] [1 2]]	1	[3.0]	[[0.70710678] [0.70710678]]

💡 Eigenvalues and eigenvectors computed for k=1, computation ends.

Variable Tracker

Variable	Start	After Step 1	After Step 4	Final
A	undefined	[[2 1] [1 2]]	[[2 1] [1 2]]	[[2 1] [1 2]]
k	undefined	undefined	1	1
vals	undefined	undefined	[3.0]	[3.0]
vecs	undefined	undefined	[[0.70710678] [0.70710678]]	[[0.70710678] [0.70710678]]

Key Moments - 3 Insights

Why do we use eigsh instead of eigs for symmetric matrices?

eigsh is optimized for symmetric or Hermitian matrices and is faster and more accurate, as shown in step 2 where we choose the solver based on matrix properties.

What does the parameter k mean in eigs/eigsh?

k is the number of eigenvalues and eigenvectors to compute, as seen in step 3 and 4 where k=1 means we get the largest eigenvalue and its vector.

Why are eigenvalues complex numbers even for real symmetric matrices?

eigs returns complex numbers by default for general matrices; for symmetric matrices, eigenvalues are real and returned as real numbers by eigsh, like 3.0 in step 4.

Visual Quiz - 3 Questions

Look at the execution table at step 4, what is the eigenvalue computed?

A [3.0]

B [2.0]

C [1.0]

D [0.0]

💡 Hint

Check the 'Eigenvalues computed' column at step 4 in the execution_table.

At which step does the code actually compute eigenvalues and eigenvectors?

A Step 2

B Step 3

C Step 4

D Step 5

💡 Hint

Look for the step where 'Compute eigenvalues and eigenvectors' happens in the execution_table.

If we change k from 1 to 2, how would the eigenvalues in the table change?

A Eigenvalues remain the same

B We get two eigenvalues instead of one

C No eigenvalues are computed

D Eigenvalues become zero

💡 Hint

Refer to the meaning of k in the key_moments and execution_table steps 3 and 4.

Concept Snapshot

Use scipy.sparse.linalg.eigs or eigsh to find eigenvalues and eigenvectors.
- eigs: for general square matrices.
- eigsh: for symmetric/Hermitian matrices (faster, more accurate).
- Parameter k controls how many eigenvalues/vectors to compute.
- Returns complex eigenvalues and eigenvectors (even if imaginary part is zero) for eigs; eigsh returns real values for symmetric matrices.
- Useful for large sparse matrices.

Full Transcript

We start by defining a matrix A. Then we choose the solver: eigs for general matrices or eigsh for symmetric ones. We call the solver with A and the number k of eigenvalues to find. The solver computes eigenvalues and eigenvectors and returns them. For example, with A = [[2,1],[1,2]] and k=1, eigsh returns the largest eigenvalue 3.0 and its eigenvector. The eigenvalues are real numbers for symmetric matrices. The parameter k controls how many eigenvalues and eigenvectors we get. Using eigsh for symmetric matrices is faster and more accurate. This process helps us analyze matrix properties in data science and engineering.