Evalues

Evalues, often written e-values or eigenvalues, are scalars λ associated with a linear transformation or square matrix A such that Av = λv for some nonzero vector v, called an eigenvector. Equivalently, they are the roots of the characteristic polynomial det(A - λI) = 0. The set of all eigenvalues of A is the spectrum of A; eigenvalues may be real or complex. If A is diagonalizable, it can be written A = PDP^{-1} with D containing its eigenvalues along the diagonal. The multiplicity of an eigenvalue can be algebraic (its multiplicity as a root) and geometric (the dimension of its eigenspace).

Computation and numerical methods: For small matrices, eigenvalues can be found by solving the characteristic polynomial, but in practice they are computed using algorithms such as the QR algorithm, power iteration, inverse iteration, or shifted versions. For large sparse matrices, iterative solvers like Lanczos or Arnoldi methods are common, often implemented in software packages such as LAPACK, ARPACK, or SciPy. Numerical accuracy depends on conditioning of A and on the eigenvalue distribution; nearly repeated or clustered eigenvalues can pose challenges.

Applications: Eigenvalues arise in stability analysis of dynamical systems, vibration modes in mechanical structures, and the spectral decomposition of symmetric matrices. They underpin principal component analysis, spectral clustering, Markov chain analysis (the dominant eigenvalue of stochastic matrices), and quantum mechanics (energy levels).

See also: eigenvectors, spectral theorem, characteristic polynomial.