Introduction
Welcome to another exciting edition of Mastering Linear Algebra! Today, we will delve into the fascinating world of Eigen Decomposition, a powerful matrix factorization technique that lies at the core of many machine learning algorithms. This concept plays a crucial role in applications like Principal Component Analysis (PCA) and dimensionality reduction.
Understanding Eigen Decomposition allows us to break down a matrix into its essential components — eigenvalues and eigenvectors. These components illuminate the underlying properties of the matrix, making it easier to comprehend and manipulate. Once you grasp this fundamental concept, you’ll be well-prepared to explore more advanced topics like Singular Value Decomposition (SVD) and matrix diagonalization. By the end of this tutorial, you’ll not only know how to calculate eigen decomposition manually but also use Python to perform the computation.
What is Eigen Decomposition?
Eigen Decomposition involves representing a square matrix A as:
Where:
- P is a matrix of eigenvectors,
- D is a diagonal matrix of eigenvalues,
- P⁻¹ is the inverse of P.
Eigenvalues describe how a linear transformation scales a vector, while eigenvectors remain unaffected by the transformation. Although not all matrices can be decomposed, those that can offer a more interpretable form of the original matrix.
Applications of Eigen Decomposition
- Principal Component Analysis (PCA): Eigen decomposition is essential for identifying principal components in PCA, enabling effective dimensionality reduction.
- Matrix Diagonalization: Eigen decomposition simplifies intricate matrix operations through diagonalization.
- Solving Systems of Linear Equations: Eigenvalues and eigenvectors offer insights into solving complex differential equations and dynamic systems.
Interested in computing eigen decomposition manually for a 2×2 matrix? Let’s get started!
Consider the matrix:
- Find the Eigenvalues:
- To determine the eigenvalues, solve the characteristic equation:
For the given matrix, expanding the determinant yields the characteristic equation:
Consequently, the eigenvalues are λ₁ = 0 and λ₂ = 5.
2. Find the Eigenvectors:
- For each eigenvalue, solve (A — λI) v = 0 to discover the corresponding eigenvector v
For λ₁ = 0:
The solution gives v₁ = [1, 2].
For λ₂ = 5:
The solution gives v₂ = [2, 1].
- Construct the Matrix P and D: Utilize the eigenvectors to create P and the eigenvalues to generate D.
Curious to see how eigen decomposition is done using Python? Let’s move on to the code implementation.
import numpy as np
# Define the matrix
A = np.array([[4, -2],
[-2, 1]])
# Perform eigen decomposition
eigenvalues, eigenvectors = np.linalg.eig(A)
print("Eigenvalues:", eigenvalues)
print("Eigenvectors:\n", eigenvectors)
Output:
Eigenvalues: [5. 0.]
Eigenvectors:
[[ 0.89442719 0.4472136 ]
[-0.4472136 0.89442719]]
Reconstructing the Original Matrix:
By leveraging the eigenvalues and eigenvectors, we can reconstruct the initial matrix A:
# Reconstruct A using P, D, and P^(-1)
P = eigenvectors
D = np.diag(eigenvalues)
P_inv = np.linalg.inv(P)
A_reconstructed = P @ D @ P_inv
print("Reconstructed Matrix A:\n", A_reconstructed)
Output:
Reconstructed Matrix A:
[[ 4. -2.]
[-2. 1.]]
Voila! We’ve successfully reconstructed the original matrix A. Eigen decomposition, with its transformative power and versatility, opens up a world of possibilities in linear algebra and machine learning. Stay tuned for more exciting adventures in the realm of matrices and vectors!