Eigenvalues and eigenvectors (eig) in SciPy - Mini Project: Build & Apply

Finding Eigenvalues and Eigenvectors with SciPy

📖 Scenario: You work as a data analyst for a company that studies vibrations in mechanical systems. You need to find the natural vibration modes of a system represented by a matrix.

🎯 Goal: Learn how to calculate eigenvalues and eigenvectors of a matrix using SciPy's eig function.

📋 What You'll Learn

Create a 2x2 matrix as a NumPy array

Import the eig function from scipy.linalg

Use eig to find eigenvalues and eigenvectors

Print the eigenvalues and eigenvectors

💡 Why This Matters

🌍 Real World

Eigenvalues and eigenvectors help engineers understand how structures vibrate and respond to forces.

💼 Career

Data scientists and engineers use eigen decomposition in signal processing, principal component analysis, and system stability analysis.

1

Create the matrix

Create a variable called matrix and set it to a 2x2 NumPy array with values [[4, 2], [1, 3]].

SciPy

import numpy as np
# Create the matrix variable as a 2x2 NumPy array
# Your code here

Hint

Use np.array to create the matrix with the exact values.

2

Import the eig function

Import the eig function from the scipy.linalg module.

SciPy

import numpy as np
matrix = np.array([[4, 2], [1, 3]])
# Import eig from scipy.linalg
# Your code here

Hint

Use from scipy.linalg import eig to import the function.

3

Calculate eigenvalues and eigenvectors

Use the eig function with the variable matrix to calculate eigenvalues and eigenvectors. Store the results in variables called eigenvalues and eigenvectors.

SciPy

import numpy as np
matrix = np.array([[4, 2], [1, 3]])
from scipy.linalg import eig
# Calculate eigenvalues and eigenvectors
# Your code here

Hint

Call eig(matrix) and unpack the result into eigenvalues and eigenvectors.

4

Print the eigenvalues and eigenvectors

Print the variables eigenvalues and eigenvectors to display the results.

SciPy

import numpy as np
matrix = np.array([[4, 2], [1, 3]])
from scipy.linalg import eig
eigenvalues, eigenvectors = eig(matrix)
# Print eigenvalues and eigenvectors
# Your code here

Hint

Use two print statements, one for eigenvalues and one for eigenvectors.

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