Lernen Implementing Eigenvectors & Eigenvalues in Python

Eigenvalues and eigenvectors are fundamental in linear algebra and arise in many data science applications — from PCA and dimensionality reduction to understanding systems dynamics and optimization.
They allow us to understand how matrices stretch or rotate vectors in space.

Defining the Matrix

# Define matrix A (square matrix)
A = np.array([[2, 1],
              [1, 2]])

This matrix represents a symmetric $2×2$ system. We're going to analyze its properties through decomposition.

Computing Eigenvalues and Eigenvectors


              12345678910111213
            
import numpy as np
from numpy.linalg import eig

# Define matrix A (square matrix)
A = np.array([[2, 1],
              [1, 2]])

# Solve for eigenvalues and eigenvectors
eigenvalues, eigenvectors = eig(A)

# Print eigenvalues and eigenvectors
print(eigenvalues)
print(eigenvectors)

NumPy's eig() computes the solutions to the equation:

A v = \lambda v

eigenvalues: a list of scalars $\lambda$ that scale eigenvectors;
eigenvectors: columns representing $v$ (directions that don’t change under transformation).

Validating Each Pair (Key Step)


              12345678910111213141516
            
import numpy as np
from numpy.linalg import eig

# Define matrix A (square matrix)
A = np.array([[2, 1],
              [1, 2]])

# Solve for eigenvalues and eigenvectors
eigenvalues, eigenvectors = eig(A)

# Verify that A @ v = λ * v for each eigenpair
for i in range(len(eigenvalues)):
    λ = eigenvalues[i]
    v = eigenvectors[:, i].reshape(-1, 1)
    print(A @ v)
    print(λ * v)

This checks if:

A v = \lambda v

The two sides should match closely, which confirms correctness. This is how we validate theoretical properties numerically.

War alles klar?

Danke für Ihr Feedback!

Abschnitt 4. Kapitel 12

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Suggested prompts:

Can you explain what eigenvalues and eigenvectors represent in simple terms?

How do I interpret the output of the eigenvalues and eigenvectors in this example?

Can you walk me through the validation step in more detail?

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Defining the Matrix

# Define matrix A (square matrix)
A = np.array([[2, 1],
              [1, 2]])

This matrix represents a symmetric $2×2$ system. We're going to analyze its properties through decomposition.

Computing Eigenvalues and Eigenvectors


              12345678910111213
            
import numpy as np
from numpy.linalg import eig

# Define matrix A (square matrix)
A = np.array([[2, 1],
              [1, 2]])

# Solve for eigenvalues and eigenvectors
eigenvalues, eigenvectors = eig(A)

# Print eigenvalues and eigenvectors
print(eigenvalues)
print(eigenvectors)

NumPy's eig() computes the solutions to the equation:

A v = \lambda v

eigenvalues: a list of scalars $\lambda$ that scale eigenvectors;
eigenvectors: columns representing $v$ (directions that don’t change under transformation).

Validating Each Pair (Key Step)


              12345678910111213141516
            
import numpy as np
from numpy.linalg import eig

# Define matrix A (square matrix)
A = np.array([[2, 1],
              [1, 2]])

# Solve for eigenvalues and eigenvectors
eigenvalues, eigenvectors = eig(A)

# Verify that A @ v = λ * v for each eigenpair
for i in range(len(eigenvalues)):
    λ = eigenvalues[i]
    v = eigenvectors[:, i].reshape(-1, 1)
    print(A @ v)
    print(λ * v)

This checks if:

A v = \lambda v

The two sides should match closely, which confirms correctness. This is how we validate theoretical properties numerically.

War alles klar?

Danke für Ihr Feedback!

Abschnitt 4. Kapitel 12

Implementing Eigenvectors & Eigenvalues in Python

Defining the Matrix

Computing Eigenvalues and Eigenvectors

Validating Each Pair (Key Step)

1. What does `np.linalg.eig(A)` return?

2. Which equation defines eigenvectors and eigenvalues?

3. What is the geometric interpretation of an eigenvector?

4. Why do we check $A v = \lambda v$ for each pair?

5. What happens if matrix A has complex eigenvalues?

Awesome!

Implementing Eigenvectors & Eigenvalues in Python

Defining the Matrix

Computing Eigenvalues and Eigenvectors

Validating Each Pair (Key Step)

1. What does `np.linalg.eig(A)` return?

2. Which equation defines eigenvectors and eigenvalues?

3. What is the geometric interpretation of an eigenvector?

4. Why do we check $A v = \lambda v$ for each pair?

5. What happens if matrix A has complex eigenvalues?

Implementing Eigenvectors & Eigenvalues in Python

Defining the Matrix

Computing Eigenvalues and Eigenvectors

Validating Each Pair (Key Step)

1. What does np.linalg.eig(A) return?

2. Which equation defines eigenvectors and eigenvalues?

3. What is the geometric interpretation of an eigenvector?

4. Why do we check Av=λvA v = \lambda vAv=λv for each pair?

5. What happens if matrix A has complex eigenvalues?

Awesome!

Implementing Eigenvectors & Eigenvalues in Python

Defining the Matrix

Computing Eigenvalues and Eigenvectors

Validating Each Pair (Key Step)

1. What does np.linalg.eig(A) return?

2. Which equation defines eigenvectors and eigenvalues?

3. What is the geometric interpretation of an eigenvector?

4. Why do we check Av=λvA v = \lambda vAv=λv for each pair?

5. What happens if matrix A has complex eigenvalues?

1. What does `np.linalg.eig(A)` return?

4. Why do we check $A v = \lambda v$ for each pair?

1. What does `np.linalg.eig(A)` return?

4. Why do we check $A v = \lambda v$ for each pair?