Eigenvalues and eigenvectors (eig) in SciPy

Introduction

Eigenvalues and eigenvectors help us understand important properties of a matrix. They show directions that don't change when the matrix acts on them, and how much they stretch or shrink.

To find main directions of data in Principal Component Analysis (PCA).

To analyze stability in systems like population growth or mechanical vibrations.

To simplify complex matrix operations by working with eigenvalues.

To understand transformations in graphics and physics.

To solve systems of linear differential equations.

Syntax

SciPy

from scipy.linalg import eig

values, vectors = eig(matrix)

matrix must be a square 2D array (same number of rows and columns).

values are the eigenvalues, and vectors are the eigenvectors as columns.

Examples

Simple diagonal matrix where eigenvalues are the diagonal elements.

SciPy

import numpy as np
from scipy.linalg import eig

A = np.array([[2, 0], [0, 3]])
values, vectors = eig(A)

Matrix representing rotation; eigenvalues are complex numbers.

SciPy

import numpy as np
from scipy.linalg import eig

B = np.array([[0, -1], [1, 0]])
values, vectors = eig(B)

Sample Program

This program calculates and prints the eigenvalues and eigenvectors of matrix M.

SciPy

import numpy as np
from scipy.linalg import eig

# Define a 2x2 matrix
M = np.array([[4, 2], [1, 3]])

# Calculate eigenvalues and eigenvectors
values, vectors = eig(M)

# Print eigenvalues
print('Eigenvalues:')
print(values)

# Print eigenvectors
print('\nEigenvectors (each column is one):')
print(vectors)

Important Notes

Eigenvalues can be complex numbers, even if the matrix has only real numbers.

Eigenvectors are normalized by default in scipy.linalg.eig.

Order of eigenvalues and eigenvectors corresponds; the first eigenvalue matches the first eigenvector column.

Summary

Eigenvalues show how much vectors stretch or shrink under a matrix.

Eigenvectors show directions that stay the same when the matrix acts on them.

Use scipy.linalg.eig to find both easily for square matrices.

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