Symmetric eigensolvers and Ritz values

For a real symmetric matrix, orthogonal similarity transforms A into QᵀAQ without changing its eigenvalues. Dense algorithms commonly reduce A to symmetric tridiagonal form using Householder transformations, then solve that structured problem using a suitable method. The original eigenvectors are recovered by applying Q to the tridiagonal eigenvectors. The key feature is preservation of both symmetry and Euclidean geometry. Computing a characteristic polynomial and finding its roots is generally a poor numerical substitute, because coefficient formation can lose the matrix problem's favorable perturbation structure.

Under a symmetric perturbation E, each ordered eigenvalue changes by at most ‖E‖₂. This absolute bound does not promise small relative error for an eigenvalue very close to zero. Eigenvectors have a different sensitivity: when eigenvalues nearly coincide, a tiny perturbation can rotate their individual eigenvectors substantially. An invariant subspace associated with a separated cluster can be better determined than its individual basis vectors. At an exact repeated eigenvalue, choosing one normalized eigenvector is not a uniquely posed output at all.

Take an orthonormal matrix V whose columns span a search space. The projected matrix T=VᵀAV is symmetric. An eigenpair of T gives a Ritz value θ and Ritz vector u=Vy. The defining Galerkin condition makes the residual Au−θu orthogonal to every search direction. This is analogous to an orthogonality condition in least squares, but the eigenproblem is different: both a scalar θ and a vector u are unknown. A Ritz value is not automatically an exact eigenvalue of A.

For any unit vector u and real θ, expanding u in an orthonormal eigenbasis proves that the residual norm is at least the distance from θ to the spectrum. Thus a small residual guarantees some true eigenvalue nearby. It does not identify which eigenvalue without separation information and does not by itself bound the angle to one selected eigenvector. For the Rayleigh quotient θ=uᵀAu, the residual is perpendicular to u. When a simple eigenvalue is separated from the others, residual-and-gap estimates can additionally control eigenvector angle.

Lanczos is the symmetric counterpart of Arnoldi: exact symmetry reduces the projected Hessenberg matrix to tridiagonal form and allows a three-term recurrence. This can make large sparse eigenproblems economical when only a few eigenvalues are required. In finite precision, orthogonality can be lost and previously converged eigenvalues may reappear as misleading copies. Reorthogonalization, restarting, and locking converged vectors address such effects at additional cost. The short exact recurrence is not a guarantee that a finite-precision basis remains orthonormal indefinitely.

Validate an eigensolver using residual norms and orthogonality checks, not only plausible-looking eigenvalues. If eigenvalues form a tight cluster, assess the subspace residual or projector rather than requiring one particular basis orientation. Specify whether accuracy is absolute or relative and whether only a spectral interval is needed. A dense all-eigenpairs computation and a sparse few-eigenpairs computation have different storage and work profiles. The most appropriate method follows from matrix structure, desired spectral information, and the reliability of its stopping certificate.