1. Eigenvalues and Eigenvectors
Av = lambda v (v nonzero)
Eigenvalue: A scalar lambda such that Av = lambda v has a nontrivial solution. Eigenvector: A nonzero vector v satisfying Av = lambda v.
Geometric meaning: eigenvectors are directions that the transformation merely scales. lambda > 0 stretches; lambda < 0 reverses; lambda = 0 collapses.
2. The Characteristic Equation
det(A - lambda I) = 0
Characteristic Polynomial: p(lambda) = det(A - lambda I), a degree-n polynomial.
2x2 shortcut: p(lambda) = lambda^2 - trace(A)*lambda + det(A). Triangular matrices: eigenvalues = diagonal entries.
3. Multiplicity
| Type | Definition | Key Fact |
|---|
| Algebraic | Multiplicity as root of p(lambda) | Sum of all = n |
| Geometric | dim(eigenspace) = dim null(A - lambda I) | 1 ≤ geo ≤ alg |
4. Diagonalization
A = PDP^{-1}, where P = [eigenvectors], D = diag(eigenvalues)
Diagonalizable iff A has n linearly independent eigenvectors (geo mult = alg mult for each eigenvalue).
n distinct eigenvalues guarantees diagonalizability. A = PDP^{-1} implies A^n = PD^nP^{-1}.
Procedure: (1) Find eigenvalues from det(A - lambda I) = 0. (2) Find eigenspaces. (3) Check total eigenvectors = n. (4) Build P and D.
5. Properties
det(A) = product of eigenvalues. trace(A) = sum of eigenvalues. lambda = 0 iff A is singular.
Complex eigenvalues of real matrices come in conjugate pairs: a + bi and a - bi.
6. Applications
Markov Chains: Stochastic matrix (columns sum to 1). Steady state q: Aq = q (eigenvector for lambda = 1, normalized).
Dynamical Systems: x_{k+1} = Ax_k. Long-term behavior determined by dominant eigenvalue. |lambda| < 1 = decay, |lambda| > 1 = growth.