Eigenvalues & Eigenvectors

Eigenvectors are directions a matrix only stretches. Learn the characteristic equation, computing eigenvalues and eigenvectors, algebraic and geometric multiplicity, diagonalisation A = PDP⁻¹, symmetric matrices and orthogonal eigenvectors, powers and exponentials of matrices, and engineering uses in vibration modes, stability, principal stresses and PCA.

The natural frequencies and mode shapes of a building, the stability of a control system, the principal stresses in a component, and Google's PageRank are all eigenvalue problems.

An analogy

When you stretch a rubber sheet, most arrows drawn on it change direction — but a few special arrows only get longer or shorter. Those are eigenvectors; the stretch factors are eigenvalues.

Finding them

Solve det(A − λI) = 0 for λ (polynomial of degree n); for each λ solve (A − λI)v = 0. Eigenvectors are defined up to scaling.

Diagonalisation

With n independent eigenvectors, A = PDP⁻¹; powers and exponentials become trivial. Repeated eigenvalues may lack enough eigenvectors (defective matrices).

Symmetric matrices

Real eigenvalues and orthogonal eigenvectors (spectral theorem): A = QΛQᵀ. Stiffness, mass, covariance and stress tensors are symmetric.

Engineering meaning

Vibration: eigenvalues ω², eigenvectors = mode shapes. Stability: Re λ < 0. Principal stresses: eigenvalues of the stress tensor. PCA: eigenvectors of covariance = main directions of variation. Large problems use iterative algorithms (power, Lanczos).

Formulas

• Definition: Av = λv
• Characteristic equation: det(A − λI) = 0
• Trace & determinant: Σλᵢ = tr A, Πλᵢ = det A
• Diagonalisation: A = PDP⁻¹ ⇒ Aᵏ = PDᵏP⁻¹, e^(At) = Pe^(Dt)P⁻¹
• Vibration: Kφ = ω²Mφ

Worked example: 2 × 2 eigenproblem

Given: A = [[4, 1], [2, 3]]

Find: Eigenvalues and eigenvectors.

• λ² − 7λ + 10 = 0 → λ = 5, 2
• λ = 5: (−1)x + y = 0 → v = (1, 1)
• λ = 2: 2x + y = 0 → v = (−1, 2)

Answer: λ = 5, 2; check tr = 7, det = 10 ✓

Worked example: Two-mass vibration

Given: m = 1 kg each, K = 100[[2, −1], [−1, 2]] N/m

Find: Natural frequencies.

• Eigenvalues of K: 100, 300
• ω = 10, 17.32 rad/s

Answer: Modes: in-phase (1, 1) at 10 rad/s; out-of-phase (1, −1) at 17.3 rad/s

Common mistakes

• Wrong: Accepting v = 0 as an eigenvector. — Right: Eigenvectors are non-zero by definition.
• Wrong: Assuming all matrices are diagonalisable. — Right: Defective matrices need Jordan forms.

Applications

• Structural dynamics: Modal analysis in FEA.
• Control & stability: Poles as eigenvalues of A.
• Data science: PCA and spectral clustering.

Key takeaways

• Av = λv; det(A − λI) = 0.
• tr = Σλ, det = Πλ.
• A = PDP⁻¹ simplifies powers and exponentials.
• Modes, stability, principal stresses, PCA.

Going further

• Power iteration: Dominant eigenvalue computation.
• Jordan form: Defective matrices.
• Generalised eigenproblems: Kφ = λMφ with Lanczos.