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.

Practice

1. What do eigenvalues represent in the context of a matrix transformation?

easy

A. The sum of all matrix elements

B. The directions that change completely under the matrix

C. How much eigenvectors stretch or shrink when the matrix is applied

D. The inverse of the matrix

Solution

1. Step 1: Understand eigenvalues meaning

   Eigenvalues measure the factor by which eigenvectors are stretched or shrunk when multiplied by the matrix.

2. Step 2: Compare options with definition

   How much eigenvectors stretch or shrink when the matrix is applied correctly describes this stretching/shrinking effect. Other options describe unrelated concepts.

3. Final Answer:

   How much eigenvectors stretch or shrink when the matrix is applied -> Option C

4. Quick Check:

   Eigenvalues = stretch/shrink factor [OK]

Hint: Eigenvalues tell how vectors scale under the matrix [OK]

Common Mistakes:

• Confusing eigenvalues with eigenvectors
• Thinking eigenvalues are directions
• Mixing eigenvalues with matrix sum

2. Which of the following is the correct way to import the function to compute eigenvalues and eigenvectors using scipy?

easy

A. from scipy.linalg import eig

B. import scipy.eigen as eig

C. from scipy.linalg import eigenvalues

D. import scipy.linalg.eig as eigen

Solution

1. Step 1: Recall scipy function for eigenvalues

   The function to compute eigenvalues and eigenvectors is named 'eig' and is in scipy.linalg.

2. Step 2: Check import syntax

   from scipy.linalg import eig correctly imports 'eig' from 'scipy.linalg'. Other options use incorrect module names or syntax.

3. Final Answer:

   from scipy.linalg import eig -> Option A

4. Quick Check:

   Correct import = from scipy.linalg import eig [OK]

Hint: Use 'from scipy.linalg import eig' to get eigenvalues [OK]

Common Mistakes:

• Using wrong module names like scipy.eigen
• Trying to import non-existent functions
• Incorrect import syntax

3. What is the output of the following code?

import numpy as np
from scipy.linalg import eig
A = np.array([[2, 0], [0, 3]])
values, vectors = eig(A)
print(values)

medium

A. [2.+0.j 3.+0.j]

B. [0.+2.j 0.+3.j]

C. [5.+0.j 0.+0.j]

D. [2 3]

Solution

1. Step 1: Understand matrix and eig function

   Matrix A is diagonal with values 2 and 3 on the diagonal. Eigenvalues of a diagonal matrix are the diagonal elements.

2. Step 2: Predict output of eig

   scipy.linalg.eig returns eigenvalues as complex numbers, so output will be [2.+0.j 3.+0.j].

3. Final Answer:

   [2.+0.j 3.+0.j] -> Option A

4. Quick Check:

   Diagonal matrix eigenvalues = diagonal elements [OK]

Hint: Diagonal matrix eigenvalues = diagonal entries as complex numbers [OK]

Common Mistakes:

• Expecting real numbers instead of complex
• Adding eigenvalues instead of listing
• Confusing eigenvalues with eigenvectors

4. The following code throws an error. What is the main issue?

import numpy as np
from scipy.linalg import eig
A = np.array([[1, 2, 3], [4, 5, 6]])
values, vectors = eig(A)

medium

A. eig function is misspelled

B. Matrix A is not square

C. Missing import for numpy

D. Vectors variable name is invalid

Solution

1. Step 1: Check matrix shape

   Matrix A has shape (2, 3), which is not square (rows != columns).

2. Step 2: Understand eig requirements

   scipy.linalg.eig requires a square matrix to compute eigenvalues and eigenvectors.

3. Final Answer:

   Matrix A is not square -> Option B

4. Quick Check:

   eig needs square matrix [OK]

Hint: Check matrix shape; eig needs square matrix [OK]

Common Mistakes:

• Ignoring matrix shape requirement
• Assuming eig works on any matrix
• Blaming variable names or imports

5. Given the matrix B = [[4, 1], [2, 3]], which of the following is a correct eigenvalue computed using scipy.linalg.eig?

hard

A. 7.0 + 0.j

B. 3.0 + 0.j

C. 1.0 + 0.j

D. 5.0 + 0.j

Solution

1. Step 1: Calculate eigenvalues of B

   Eigenvalues satisfy det(B - λI) = 0. For B=[[4,1],[2,3]], characteristic polynomial is (4-λ)(3-λ)-2=0 -> λ² -7λ +10=0.

2. Step 2: Solve quadratic equation

   Roots are λ = (7 ± sqrt(49 - 40))/2 = (7 ± 3)/2 -> λ=5 or λ=2.

3. Step 3: Match options

   5.0 + 0.j is 5.0 + 0.j, which matches one eigenvalue. Others do not match.

4. Final Answer:

   5.0 + 0.j -> Option D

5. Quick Check:

   Eigenvalues = 5 and 2 [OK]

Hint: Solve characteristic polynomial for eigenvalues [OK]

Common Mistakes:

• Miscomputing characteristic polynomial
• Ignoring complex number format
• Choosing wrong root from quadratic