1.15 Eigenvalues and Eigenvectors

AHL 1.15: EIGENVALUES & EIGENVECTORS

📌 Key terms & quick notes

Term Definition / Notes
Eigenvalue (λ) Scalar λ for which a non-zero vector v exists with A v = λ v. Solve via characteristic polynomial det(A − λ I) = 0.
Eigenvector (v) Non-zero vector direction preserved by A (only scaled). Eigenspace = nullspace of (A − λ I).
Characteristic polynomial p(λ) = det(A − λ I). For 2×2: p(λ) = λ2 − (tr A) λ + det A.
Trace / Determinant tr A = sum diag entries = λ1 + λ2. det A = product λ1 × λ2.
Algebraic / Geometric multiplicity Algebraic = root multiplicity in p(λ). Geometric = dimension of eigenspace (≤ algebraic).
Diagonalizable A = P D P−1 with independent eigenvectors as columns of P. Distinct eigenvalues ⇒ diagonalizable (2×2).

🔢 GDC Tips

• Use matrix → eigen routines to compute eigenvalues and eigenvectors quickly.
• For powers An, diagonalize (A = P D P−1) then Dn just raises diagonal entries.
• Verify results by computing A v − λ v; numerical residual should be ≈ 0. Use rational/exact mode when available.

📌 Geometric intuition

• Direction preservation: eigenvectors are directions that A maps onto the same line (scaled by λ). They are the “axes” along which the linear transformation purely stretches/compresses (or flips).

• Magnitude effect: |λ| < 1 ⇒ decay along that mode; |λ| > 1 ⇒ growth. λ negative ⇒ flip sign (mirror) plus scaling. Complex λ give rotation+scaling.

• Modes: decomposing an initial state into eigen-components isolates independent behaviours: each component evolves as λt (discrete) or eλ t (continuous).

📌 Step-by-step computations & meaning

  1. Form A − λI and compute det(A − λI): detects λ that make matrix singular → non-trivial solutions for v.
  2. Solve p(λ)=0: yields the scaling factors (modes) — algebraic roots that indicate possible behaviours.
  3. Solve (A − λI) v = 0: finds eigendirections v (geometric meaning: preserved lines).
  4. Diagonalize: A = P D P−1 means change to eigenbasis where A acts as scaling (D). This simplifies repeated application and interpretation.
  5. Use decomposition to compute An: Dn = diag(λin) so long-term behaviour dominated by eigenvalues with largest |λ|.

📌 Holistic example — two-town transition

Model: xt = [At, Bt]T, At+1/Bt+1 = A × xt with

A = [[0.90, 0.20], [0.10, 0.80]]

  1. Characteristic: p(λ) = λ2 − (tr A) λ + det A. Here tr A = 1.70; det A = 0.70 → p(λ) = λ2 − 1.70λ + 0.70.
  2. Solve → λ1=1, λ2=0.70. Interpretation: λ=1 gives steady-state (persistent mode); 0.70 decays (transient).
  3. Eigenvector for λ=1 (solve (A−I)v=0) gives long-term population fractions; initial conditions different only affect transient coefficients.

Conclusion: eigen-analysis separates steady distribution (end behaviour) from decaying components (how fast equilibrium is reached).

🧮 IA Tips

Use a simple transition dataset (migration, market share) so you can compute eigenpairs and perform sensitivity (±δ changes). Explain model assumptions and validate with GDC and hand-calculations. Discuss limitations and measurement error impact on eigenvalues/eigenvectors.

📌 Stability & continuous analogues

• Discrete: xt+1 = A xt stable if all |λ| < 1; dominated by largest |λ| otherwise.
• Continuous: x'(t) = B x(t) solved via eBt; eigenvalues of B (real parts) govern stability (Re(λ) < 0 stable).

📌 Numerical cautions & exam checks

  • Close eigenvalues: ill-conditioning — small A changes alter eigenvectors strongly.
  • Repeated eigenvalue: check geometric multiplicity; if less than algebraic, not diagonalizable — show dimension of nullspace.
  • Floating point: prefer rational/exact results when possible; state rounding explicitly where needed.
  • Verification: always compute A v − λ v (or residual) to validate eigenpairs numerically.

🧠 Examiner Tip

Show determinant computation (det(A − λI)), factor the characteristic polynomial, and solve (A − λI)v = 0. If eigenvalues repeat, compute nullspace dimension and explain diagonalizability. When using the GDC, include a short algebraic justification to secure method marks.

🔍 TOK Perspective

How do idealizations (linearity) shape what we consider knowledge? Discuss assumptions, model domains and ethical consequences of applying simplified linear models to policy decisions.

🌍 Real-World Connection

Markov chains, population transitions, Google’s PageRank, vibration modes in engineering — all use eigen-analysis to separate persistent modes from transients and to identify dominant behaviour.

📌 Glossary & symbols

Symbol Meaning
λ Eigenvalue (scalar)
v Eigenvector (column)
P Matrix of eigenvectors (columns)
D Diagonal matrix of eigenvalues
An Matrix power — use diagonalisation for efficient computation

📝 Exam Strategy

Practice forming det(A − λI), solving polynomials, and computing eigenspaces. If using GDC, display the algebraic steps too (polynomial factorisation, nullspace dimension). When eigenvalues repeat, explicitly show geometric multiplicity check.