Linear Algebra

For a square matrix A, a non-zero vector v is an eigenvector with corresponding eigenvalue λ if:

A v = λ v

In words: A's action on v is just scaling (no direction change).

Finding eigenvalues: solve the characteristic equation det(A − λI) = 0.

Finding eigenvectors: for each λ, solve (A − λI)v = 0 (null space).

Properties of eigenvalues:

• Sum of eigenvalues = trace of A (sum of diagonal entries).
• Product of eigenvalues = det(A).
• A is invertible ⟺ no eigenvalue is 0.
• Eigenvalues of A^k are λ^k (with same eigenvectors).
• Eigenvalues of A⁻¹ are 1/λ (with same eigenvectors).
• Triangular matrix: eigenvalues are the diagonal entries.
• Symmetric matrix: all eigenvalues are real.
• Orthogonal matrix: |λ| = 1 for all eigenvalues.

Diagonalization. If A has n linearly independent eigenvectors:

A = PDP⁻¹

where P's columns are eigenvectors and D is diagonal with eigenvalues. Then A^k = PD^k P⁻¹ — exponentiating becomes trivial.

Not every matrix is diagonalizable. Required: algebraic multiplicity = geometric multiplicity for each eigenvalue.

Worked example.

A = [[2, 1], [1, 2]]. Find eigenvalues and eigenvectors.

det(A − λI) = (2−λ)² − 1 = λ² − 4λ + 3 = 0 → λ = 1, 3.

For λ = 1: (A − I)v = [[1,1],[1,1]]v = 0 → v = (1, −1).
For λ = 3: (A − 3I)v = [[−1,1],[1,−1]]v = 0 → v = (1, 1).

Trace check: 2 + 2 = 4 = 1 + 3. ✓
Det check: 4 − 1 = 3 = 1 × 3. ✓

Applications (why eigenvalues matter beyond GATE):

1. Principal Component Analysis (PCA) — eigenvectors of covariance matrix point in directions of maximum variance.
2. PageRank — Google's algorithm finds the principal eigenvector of the web's link matrix.
3. Vibration modes in mechanical / civil engineering.
4. Quantum mechanics — energy levels are eigenvalues of the Hamiltonian.
5. Markov chains — stationary distribution = left eigenvector of transition matrix with eigenvalue 1.

Cayley-Hamilton theorem: every square matrix satisfies its own characteristic equation. For 2×2: A² − (tr A) A + (det A) I = 0.

Useful trick: lets you express A⁻¹ in terms of A: rearranging Cayley-Hamilton, A⁻¹ = [(tr A) I − A] / det A.