↗️ Eigenvalues — Linear Transformations & Principal Axes

When tr(A)² − 4·det(A) < 0, eigenvalues are complex conjugates α ± βi. No real eigenvectors exist; instead the transformation combines rotation with scaling. The modulus |λ| = √(α² + β²) tells you whether points spiral inward (|λ| < 1), outward (|λ| > 1), or trace ellipses (|λ| = 1). Pure rotation matrices have eigenvalues on the unit circle with |λ| = 1.

For any square matrix, the sum of all eigenvalues equals the trace and their product equals the determinant. For a 2×2 matrix: λ₁ + λ₂ = tr(A) and λ₁·λ₂ = det(A). These are Vieta's formulas applied to the characteristic polynomial and give quick sanity checks — if det(A) = 0 then at least one eigenvalue is zero, meaning the transformation collapses a dimension.

A matrix is defective if it has fewer linearly independent eigenvectors than its size. For a 2×2 matrix this occurs when the characteristic polynomial has a repeated root but only one eigenvector direction. The classic example is [[2,1],[0,2]]: eigenvalue 2 (double), but only the eigenvector [1,0]. Defective matrices cannot be diagonalised; they require a Jordan normal form instead.

PCA finds orthogonal directions of maximum variance in a data set. It eigendecomposes the covariance matrix C = (1/n)·XᵀX: the eigenvectors are the principal components (directions of variance), and the eigenvalues quantify exactly how much variance each component captures. Sorting by descending eigenvalue lets you retain only the top k components and still explain most of the data — this is the basis of face recognition, data visualisation, and noise reduction.

For a discrete system xₙ₊₁ = A·xₙ, stability requires all eigenvalue magnitudes |λᵢ| < 1. If any |λᵢ| > 1, the system grows without bound (unstable). For continuous systems ẋ = Ax, the criterion shifts to Re(λᵢ) < 0 for all eigenvalues. These facts underpin the Lyapunov stability theory used in robotics, aircraft autopilots, and economic models.

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial. If p(λ) = det(A − λI) then p(A) = 0. For a 2×2 matrix this means A² − tr(A)·A + det(A)·I = 0. The theorem is used in control theory to express high powers of A in terms of lower powers, simplifying the analysis of state-space models.

The spectral radius ρ(A) = max|λᵢ| is the largest absolute eigenvalue. It controls the long-term growth rate of iterative maps — repeated multiplication by A amplifies vectors by approximately ρ(A)ⁿ after n steps. In numerical methods, ρ(A) < 1 guarantees convergence of iterative solvers like Jacobi or Gauss-Seidel. The Google PageRank power iteration converges because the link matrix has ρ = 1 with a unique dominant eigenvector.

Yes. If λ = 0 is an eigenvalue, then Av = 0 for some non-zero v, meaning det(A) = 0 and the matrix is singular (not invertible). The null space of A is spanned by the zero-eigenvalue eigenvectors. In geometric terms the transformation collapses at least one dimension — a 2D matrix with a zero eigenvalue squashes the plane onto a line or a point.