Eigenvalue decomposition

Eigenvalue decomposition (or spectral decomposition) is the factorization of a linear operator into its intrinsic scaling directions. For a square matrix, it asks: in which directions does the transformation act as mere scaling, without rotation or shearing? The answer is given by eigenvectors (the directions) and eigenvalues (the scale factors).

Formally, for a matrix A, an eigenvector v and eigenvalue λ satisfy Av = λv. The eigenvectors span the directions that A preserves; the eigenvalues encode how much stretching or compression occurs along those directions. When A is symmetric (or Hermitian in the complex case), the spectral theorem guarantees real eigenvalues and orthogonal eigenvectors — a complete, uncoupled description of the operator's action.

Eigenvalue decomposition is the analytical heart of principal component analysis, stability analysis, and quantum measurement. It reveals the intrinsic coordinate system in which a complex transformation becomes simple scaling. Without it, linear algebra would be merely arithmetic; with it, linear algebra becomes geometry.

The limitation is clear: not all matrices admit full eigenvalue decomposition. Defective matrices — those with repeated eigenvalues but insufficient eigenvectors — resist diagonalization. This is why the singular value decomposition is more general: it works for all matrices, at the cost of using two different orthonormal bases rather than one. Eigenvalue decomposition is the special case where the two bases coincide.