Eigenproblems

An eigenproblem, in linear algebra, asks for scalars and vectors that satisfy A x = λ x for a square matrix A or for a linear operator T on a vector space, where λ is an eigenvalue and x is a corresponding eigenvector. In the generalized eigenproblem, A x = λ B x, with B invertible, the eigenvalues are found relative to the pair (A, B). In functional analysis, eigenproblems take the form T f = λ f, where T is a linear operator and f is an eigenfunction.

Key properties include that eigenvalues may be real or complex depending on A, and eigenvectors corresponding to distinct eigenvalues are linearly independent. The characteristic polynomial det(A − λI) = 0 determines the eigenvalues, with algebraic multiplicity and geometric multiplicity describing their roots and independent eigenvectors. A matrix is diagonalizable if it has a full set of linearly independent eigenvectors; not all matrices possess this property.

Computationally, analytical solutions exist for small matrices, while large or sparse problems require numerical methods. Common techniques include the power iteration, inverse iteration, and the QR algorithm; for large sparse problems, Arnoldi or Lanczos methods are used to approximate a few eigenpairs. Generalized eigenproblems are solved by algorithms such as the QZ method. Numerical stability, conditioning, and perturbation effects are important considerations in practical calculations.

Applications are widespread. Eigenvectors and eigenvalues underpin principal component analysis in statistics, modal analysis in engineering, and the study of natural vibration modes. In quantum mechanics, eigenvalues of observables correspond to possible measurement outcomes. Markov chains use stationary eigenvectors of transition matrices, and differential equations often lead to eigenvalue problems through Sturm–Liouville theory.