Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra with deep applications in machine learning, such as PCA (Principal Component Analysis), dimensionality reduction, and stability analysis of algorithms.

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Definition

For a square matrix $A\in {\mathbb{R}}^{n\times n}$, a nonzero vector $v$ is an eigenvector if:

$$Av=\lambda v$$

where $\lambda$ is the corresponding eigenvalue.

• $v$ gives a direction that is unchanged (up to scaling) by $A$.
• $\lambda$ tells how much the vector is stretched or shrunk.

Explanation of Eigenvalues & Eigenvectors

Think of a matrix $A$ as a transformation. Most vectors change direction when transformed, but eigenvectors keep their direction, only scaling by $\lambda$.

• In ML: PCA finds eigenvectors of the covariance matrix → principal directions of data variance.

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Mini Example

$$A=\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]$$

If $v=[1,0{]}^T$, then:

$$Av=\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}2\\ 0\end{array}\right]=2v$$

So $v$ is an eigenvector with eigenvalue $\lambda =2$.

Similarly, $[0,1{]}^T$ is an eigenvector with eigenvalue $\lambda =3$.

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Characteristic Equation

Eigenvalues are found by solving:

$$det(A-\lambda I)=0$$

This yields an $n$-degree polynomial in $\lambda$. Each solution is an eigenvalue.

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Hands-on with Python and Rust

Python

import numpy as np

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

# Eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(A)

print("Eigenvalues:", eigenvalues)
print("Eigenvectors:\n", eigenvectors)

Rust

use ndarray::array;
use ndarray::Array2;
use ndarray_linalg::Eig;

fn main() {
    let a: Array2<f64> = array![
        [2.0, 0.0],
        [0.0, 3.0]
    ];

    // Eigen decomposition
    let (eigenvalues, eigenvectors) = a.eig().unwrap();

    println!("Eigenvalues: {:?}", eigenvalues);
    println!("Eigenvectors:\n{:?}", eigenvectors);
}

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Connection to ML

• PCA → uses eigenvectors of covariance matrix to find directions of maximum variance.
• Spectral clustering → uses eigenvalues of Laplacian matrices.
• Stability analysis → eigenvalues determine convergence rates of iterative methods.

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Next Steps

Continue to Singular Value Decomposition (SVD).