Eigenvalues and eigenvectors (eig) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of eigenvalues and eigenvectors calculation

What is the output of the following code that calculates eigenvalues and eigenvectors of a matrix using scipy.linalg.eig?

SciPy

import numpy as np
from scipy.linalg import eig

A = np.array([[2, 1], [1, 2]])
eigenvalues, eigenvectors = eig(A)
print(np.round(eigenvalues, 2))
print(np.round(eigenvectors, 2))

A

[1.+0.j 3.+0.j]
[[ 0.71 -0.71]
 [ 0.71  0.71]]

B

[3.+0.j 1.+0.j]
[[ 0.71  0.71]
 [-0.71  0.71]]

C

]]17.0  17.0-[ 
]17.0  17.0 [[
]j.0+.1 j.0+.3[

D

[1.+0.j 3.+0.j]
[[ 0.71  0.71]
 [-0.71  0.71]]

💡 Hint

Remember eigenvalues are sorted in the order returned by eig, and eigenvectors correspond column-wise.

The matrix [[2,1],[1,2]] has eigenvalues 3 and 1. The eigenvector for eigenvalue 3 is [0.71, 0.71] and for 1 is [-0.71, 0.71]. The output shows eigenvalues as complex numbers with zero imaginary part.

❓ data_output

intermediate

Number of eigenvalues for a non-square matrix

What happens when you try to compute eigenvalues of a non-square matrix using scipy.linalg.eig? How many eigenvalues are returned?

SciPy

import numpy as np
from scipy.linalg import eig

B = np.array([[1, 2, 3], [4, 5, 6]])
eigenvalues, eigenvectors = eig(B)

A Raises a ValueError because the matrix is not square

B Returns an empty array with zero eigenvalues

C Returns 3 eigenvalues corresponding to the number of columns

D Returns 2 eigenvalues corresponding to the number of rows

💡 Hint

Eigenvalues are defined only for square matrices.

The eig function requires a square matrix. Passing a non-square matrix raises a ValueError indicating the input must be square.

🧠 Conceptual

advanced

Interpretation of complex eigenvalues

A real 2x2 matrix has eigenvalues 1+2j and 1-2j. What does this imply about the matrix's behavior?

A The matrix is diagonal with real entries

B The matrix is symmetric and positive definite

C The matrix has only real eigenvectors

D The matrix represents a rotation combined with scaling

💡 Hint

Complex conjugate eigenvalues often indicate rotation in 2D space.

Complex conjugate eigenvalues with nonzero imaginary parts indicate the matrix performs a rotation combined with scaling in the plane.

🔧 Debug

advanced

Identify the error in eigenvalue calculation code

What error does the following code produce and why? import numpy as np from scipy.linalg import eig C = np.array([[1, 2], [3, 4], [5, 6]]) eigenvalues, eigenvectors = eig(C)

A IndexError: index out of bounds

B TypeError: too many values to unpack

C ValueError: expected square matrix

D No error, runs successfully

💡 Hint

Check the shape of the matrix passed to eig.

The matrix C is 3x2, not square. eig requires a square matrix and raises ValueError.

🚀 Application

expert

Using eigenvectors to diagonalize a matrix

Given a symmetric matrix M, which code snippet correctly diagonalizes M using its eigenvectors and eigenvalues?

SciPy

import numpy as np
from scipy.linalg import eig

M = np.array([[4, 1], [1, 3]])

A

vals, vecs = eig(M)
D = np.diag(vals)
M_reconstructed = vecs @ D @ np.linalg.inv(vecs)

B

vals, vecs = eig(M)
D = np.diag(vals)
M_reconstructed = vecs @ D @ vecs.T

C

vals, vecs = eig(M)
D = np.diag(vals)
M_reconstructed = vecs.T @ D @ vecs

D

vals, vecs = eig(M)
D = np.diag(vals)
M_reconstructed = np.linalg.inv(vecs) @ D @ vecs

💡 Hint

Diagonalization formula: M = V D V^{-1} where V columns are eigenvectors.

For diagonalization, M = V D V^{-1} where V is eigenvector matrix and D diagonal eigenvalue matrix. Since M is symmetric, eig returns eigenvectors that are orthogonal but eig does not guarantee orthonormality, so inverse is needed, not transpose.