Eigenvalues and eigenvectors (eig) in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the function that calculates eigenvalues and eigenvectors.

SciPy

from scipy.linalg import [1]

matrix = [[2, 0], [0, 3]]
values, vectors = [1](matrix)

Drag options to blanks, or click blank then click option'

Aeig

Binv

Cdet

Dsolve

Attempts:

3 left

💡 Hint

Common Mistakes

Using functions like inv or det instead of eig.

Not importing the function before using it.

The eig function from scipy.linalg computes eigenvalues and eigenvectors of a matrix.

2fill in blank

medium

Complete the code to create a 2x2 matrix for eigenvalue calculation.

SciPy

import numpy as np

matrix = np.array([1])
values, vectors = eig(matrix)

Drag options to blanks, or click blank then click option'

A[[1, 2], [3, 4]]

B[1, 2, 3, 4]

C[[1, 2, 3], [4, 5, 6]]

D[[1, 2], [3]]

Attempts:

3 left

💡 Hint

Common Mistakes

Using a 1D list instead of a 2D array.

Creating matrices with inconsistent row lengths.

The matrix must be a 2x2 array for this example. Option A correctly defines a 2x2 matrix.

3fill in blank

hard

Fix the error in the code to correctly compute eigenvalues and eigenvectors.

SciPy

from scipy.linalg import eig

matrix = [[4, 1], [2, 3]]
values, vectors = eig([1])

Drag options to blanks, or click blank then click option'

Amatrix.tolist()

B[4, 1, 2, 3]

Cmatrix

Dnp.array(matrix)

Attempts:

3 left

💡 Hint

Common Mistakes

Passing a flat list instead of a 2D matrix.

Using undefined variables or wrong data types.

The variable matrix is already a list of lists, which eig accepts. Passing it directly works.

4fill in blank

hard

Fill both blanks to create a dictionary of eigenvalues and their corresponding eigenvectors.

SciPy

values, vectors = eig(matrix)
eigen_dict = {values[[1]]: vectors[:, [2]] for i in range(len(values))}

Drag options to blanks, or click blank then click option'

Ai

B0

C1

Dlen(values)

Attempts:

3 left

💡 Hint

Common Mistakes

Using fixed indices instead of the loop variable.

Mixing up row and column indexing for vectors.

Use the loop variable i to index both eigenvalues and eigenvectors correctly.

5fill in blank

hard

Fill all three blanks to compute eigenvalues, eigenvectors, and print the first eigenvalue and its vector.

SciPy

from scipy.linalg import [1]
import numpy as np

matrix = np.array([[5, 4], [1, 2]])
values, vectors = [2](matrix)
print('First eigenvalue:', values[[3]])
print('First eigenvector:', vectors[:, 0])

Drag options to blanks, or click blank then click option'

Aeig

Beigvals

C0

Attempts:

3 left

💡 Hint

Common Mistakes

Using eigvals which returns only eigenvalues, not vectors.

Using wrong index for the first eigenvalue.

Import and use eig to get eigenvalues and eigenvectors. The first eigenvalue is at index 0.

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