Eigenvalues & Eigenvectors

Eigenvalues and Eigenvectors

An eigenvector $mathbf{v}$ of a matrix $mathbf{A}$ is a vector that only gets scaled (not rotated) when multiplied by $mathbf{A}$:

$mathbf{A} cdot mathbf{v} = lambda cdot mathbf{v}$

The scalar $lambda$ (lambda) is the eigenvalue — it tells you how much the eigenvector is stretched.

The Characteristic Equation

Eigenvalues satisfy $det(mathbf{A} - lambda I) = 0$. For a $2 imes 2$ matrix this yields:

$lambda^2 - ext{trace}(mathbf{A}) cdot lambda + det(mathbf{A}) = 0$

Using the quadratic formula:

import math

def eigenvalues_2x2(A):
    tr = A[0][0] + A[1][1]          # trace
    d = A[0][0]*A[1][1] - A[0][1]*A[1][0]   # determinant
    disc = tr**2 - 4*d
    l1 = (tr - math.sqrt(disc)) / 2
    l2 = (tr + math.sqrt(disc)) / 2
    return sorted([int(round(l1)), int(round(l2))])

A = [[3, 0], [0, 5]]
print(eigenvalues_2x2(A))   # [3, 5]

Diagonal Matrices

For diagonal matrices, the eigenvalues are just the diagonal entries.

Why Eigenvalues Matter

• PCA (Principal Component Analysis) — eigenvectors of the covariance matrix give the principal components; eigenvalues give the variance explained
• Google PageRank — the web's link structure is a matrix; the PageRank vector is its dominant eigenvector
• Stability analysis — eigenvalues determine whether a dynamical system converges or diverges

Your Task

Implement sorted_eigenvalues(A) that returns the eigenvalues of a $2 imes 2$ matrix $mathbf{A}$ sorted in ascending order, as a list of integers.