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Module 6: Quick Reference

Eigenvalues and Eigenvectors -- Linear Algebra

Core Definitions

Av = lambda v (v nonzero) | det(A - lambda I) = 0
2x2: p(lambda) = lambda^2 - trace(A)*lambda + det(A)

Finding Eigenvalues & Eigenvectors

Step 1: Eigenvalues

Solve det(A - lambda I) = 0

Step 2: Eigenvectors

Solve (A - lambda I)v = 0

Multiplicity

Algebraic

Multiplicity as root of p(lambda)

Geometric

dim null(A - lambda I)

1 ≤ geometric ≤ algebraic (always)

Diagonalization

A = PDP^{-1} | P = [v1 v2 ... vn] | D = diag(lambda_1, ..., lambda_n)
A^n = PD^nP^{-1}
Diagonalizable iff n independent eigenvectors exist

Properties & Applications

det & trace

det(A) = product of eigenvalues

trace(A) = sum of eigenvalues

Stability

|lambda| < 1: decay

|lambda| > 1: growth

|lambda| = 1: persist

Steady State (Markov)

Solve Aq = q, normalize q so entries sum to 1