Вивчайте Eigenvalues and Eigenvectors

Свайпніть щоб показати меню

Definition

An eigenvector of a matrix is a nonzero vector whose direction remains unchanged when a linear transformation (represented by the matrix) is applied to it; only its length is scaled. The amount of scaling is given by the corresponding eigenvalue.

For covariance matrix $\Sigma$ , eigenvectors point in the directions of maximum variance, and eigenvalues tell you how much variance is in those directions.

Mathematically, for matrix $A$ , eigenvector $v$ and eigenvalue $λ$ :

A v = \lambda v

In PCA, the eigenvectors of the covariance matrix are the principal axes, and the eigenvalues are the variances along those axes.


              12345678910111213
            
import numpy as np

# Using the covariance matrix from the previous code
X = np.array([[2.5, 2.4],
              [0.5, 0.7],
              [2.2, 2.9]])
X_centered = X - np.mean(X, axis=0)
cov_matrix = (X_centered.T @ X_centered) / X_centered.shape[0]

# Compute eigenvalues and eigenvectors
values, vectors = np.linalg.eig(cov_matrix)
print("Eigenvalues:", values)
print("Eigenvectors:\n", vectors)

Note

The eigenvector with the largest eigenvalue points in the direction of greatest variance in the data. This is the first principal component.

Все було зрозуміло?

Дякуємо за ваш відгук!

Секція 1. Розділ 6

Запитати АІ

Запитайте про що завгодно або спробуйте одне із запропонованих запитань, щоб почати наш чат

Секція 1. Розділ 6