Eigenvalue problem

An eigenvalue problem concerns the determination of nonzero vectors whose directions are preserved by a linear transformation. For a linear operator (A) on a vector space, a scalar (\lambda) and a nonzero vector (v) satisfy the eigenvalue equation when

[ Av=\lambda v. ]

The scalar (\lambda) is an eigenvalue, while (v) is a corresponding eigenvector. In finite dimensions, the problem is represented by a square matrix after a basis has been selected. In infinite-dimensional settings, the same formulation applies to operators on function spaces, although domain restrictions, convergence, and the distinction between discrete and continuous spectra become essential.

Eigenvalue problems organize information about invariant directions and characteristic rates of change. Their mathematical role derives from the fact that many transformations become diagonal, or nearly diagonal, when expressed in coordinates adapted to their eigenvectors. This structure links eigenvalue analysis to differential equations, stability theory, quantum mechanics, and the numerical study of large matrices.

Finite-dimensional formulation

Let (A) be an (n\times n) matrix over a field (\mathbb F). The equation (Av=\lambda v) is equivalent to

[ (A-\lambda I)v=0, ]

where (I) denotes the identity matrix. A nonzero solution exists precisely when (A-\lambda I) is singular, which gives the characteristic equation

[ \det(A-\lambda I)=0. ]

The polynomial (p_A(\lambda)=\det(A-\lambda I)) is the characteristic polynomial of (A). Over the complex numbers it has (n) roots when multiplicities are counted, as follows from the fundamental theorem of algebra. A real matrix may possess nonreal eigenvalues, which then occur in complex-conjugate pairs.

For a fixed eigenvalue (\lambda), the associated eigenspace is

[ E_\lambda=\ker(A-\lambda I). ]

Its dimension is the geometric multiplicity of (\lambda). The multiplicity of the same root in the characteristic polynomial is its algebraic multiplicity. Geometric multiplicity never exceeds algebraic multiplicity, and equality for every eigenvalue characterizes diagonalizability over a field containing the complete spectrum.

A matrix (A) is diagonalizable when an invertible matrix (S) exists such that

[ S^{-1}AS=D, ]

with (D) diagonal. The columns of (S) are eigenvectors, while the diagonal entries of (D) are their corresponding eigenvalues. When a matrix lacks enough linearly independent eigenvectors, its structure is instead described by generalized eigenvectors and the Jordan normal form.

Left eigenvectors satisfy

[ w^{}A=\lambda w^{}, ]

where (w^{*}) is the conjugate transpose of (w). For nonnormal matrices, left and right eigenvectors need not coincide or form orthogonal families. Their relative orientation influences eigenvalue sensitivity and the conditioning of spectral decompositions.

Symmetric and normal problems

The eigenvalue problem has particularly rigid structure for Hermitian matrices. If (A=A^{*}), every eigenvalue is real, and eigenvectors belonging to distinct eigenvalues are orthogonal. The finite-dimensional spectral theorem therefore gives a unitary matrix (Q) and a real diagonal matrix (\Lambda) satisfying

[ A=Q\Lambda Q^{*}. ]

Real symmetric matrices admit the corresponding decomposition with an orthogonal matrix. More generally, a complex matrix is unitarily diagonalizable exactly when it is normal, meaning that (A^{}A=AA^{}).

For Hermitian matrices, the eigenvalues can also be characterized without explicitly solving the characteristic equation. The Rayleigh quotient,

[ R_A(x)=\frac{x^{}Ax}{x^{}x}, ]

takes values between the smallest and largest eigenvalues. The extremal values of this quotient are the extremal eigenvalues, and the broader min–max theorem characterizes each ordered eigenvalue through optimization over subspaces. These variational descriptions connect matrix spectra with constrained quadratic optimization and with energy principles in operator theory.

Positive definiteness is likewise spectral. A Hermitian matrix is positive definite precisely when all its eigenvalues are positive. This equivalence explains why eigenvalues determine the curvature of quadratic forms and the local classification of stationary points through the Hessian matrix.

Generalized eigenvalue problems

A generalized eigenvalue problem has the form

[ Av=\lambda Bv, ]

for square matrices (A) and (B). When (B) is invertible, the equation can formally be written as (B^{-1}Av=\lambda v), although the pair ((A,B)) retains structural information that may be obscured by explicitly forming (B^{-1}A). The associated characteristic equation is

[ \det(A-\lambda B)=0. ]

If (B) is singular, the matrix pencil (A-\lambda B) may contain infinite eigenvalues or singular components. The Kronecker canonical form describes the complete algebraic structure of a general matrix pencil, while regular pencils admit a spectral treatment closer to that of ordinary matrices.

Generalized symmetric problems arise when both matrices are Hermitian and (B) is positive definite. Their eigenvalues are real, and their eigenvectors can be selected to satisfy (B)-orthogonality rather than ordinary Euclidean orthogonality. Such problems occur naturally after variational equations are represented in a nonorthonormal basis.

Operator spectra

For an operator (T) on an infinite-dimensional space, eigenvalues form only part of the spectrum. A scalar (\lambda) belongs to the spectrum when (T-\lambda I) fails to possess an everywhere-defined bounded inverse. This failure may occur even if the equation (Tv=\lambda v) has no nonzero solution.

The spectrum is commonly divided into point, continuous, and residual components according to the injectivity, range, and density properties of (T-\lambda I). The point spectrum consists of ordinary eigenvalues. Continuous spectral values arise when the operator is injective and has dense range but lacks a bounded inverse, while the residual spectrum reflects failure of density in the range.

For compact operators on an infinite-dimensional Banach space, every nonzero spectral value is an eigenvalue of finite multiplicity, and zero is the only possible accumulation point. Compact self-adjoint operators on Hilbert spaces consequently possess an orthonormal eigenvector expansion analogous to the diagonalization of Hermitian matrices. This result underlies many eigenvalue formulations for integral equations and boundary-value problems.

A differential-operator eigenvalue problem often includes boundary conditions. A representative Sturm–Liouville problem is

[ -\frac{d}{dx}\left(p(x)\frac{dy}{dx}\right)+q(x)y =\lambda w(x)y, ]

together with endpoint conditions that define the operator domain. Under standard self-adjoint assumptions, its eigenvalues are real and its eigenfunctions are orthogonal with respect to the weight (w). The resulting expansions generalize Fourier series and provide modal descriptions of boundary-dependent systems.

Perturbation and conditioning

If a matrix is altered by a perturbation (E), its eigenvalues and invariant subspaces also change. For Hermitian matrices, Hermann Weyl’s perturbation result bounds the displacement of each ordered eigenvalue by the operator norm of (E). Chandler Davis and William Kahan developed corresponding bounds for changes in invariant subspaces, with the separation between spectral clusters controlling the resulting rotation.

Nonnormal matrices can behave differently because their eigenvectors may be nearly linearly dependent. The Bauer–Fike theorem states that, for a diagonalizable matrix (A=SDS^{-1}), every eigenvalue of (A+E) lies near an eigenvalue of (A), with the bound enlarged by the condition number of (S). Thus small matrix perturbations can produce comparatively large spectral changes even when the unperturbed eigenvalues are distinct.

The pseudospectrum records this sensitivity by including scalars that become eigenvalues under perturbations of a prescribed magnitude. It also characterizes points where the resolvent ((A-\lambda I)^{-1}) has a large norm. For normal matrices, pseudospectral sets are neighborhoods of the spectrum, whereas nonnormal matrices can produce regions extending far beyond such neighborhoods.

Numerical treatment

Direct expansion of the characteristic polynomial is generally unsuitable for floating-point computation because its coefficients may be poorly conditioned and because root finding introduces a second sensitive problem. Modern numerical methods instead transform the matrix through stable similarity operations or construct lower-dimensional subspaces that approximate selected spectral components.

For dense matrices, reduction to Hessenberg form precedes the QR algorithm. The QR iteration applies orthogonal or unitary transformations and converges toward a triangular Schur decomposition, whose diagonal contains the eigenvalues. Symmetric matrices reduce to tridiagonal form, allowing their real spectral structure to be exploited more directly.

For large sparse matrices, Krylov-subspace methods represent the action of (A) on spaces of the form

[ \mathcal K_m(A,v) =\operatorname{span}{v,Av,A^2v,\ldots,A^{m-1}v}. ]

The Arnoldi iteration constructs an orthonormal basis for this space and produces a smaller Hessenberg matrix whose eigenvalues approximate those of (A). The Lanczos algorithm is the Hermitian specialization, in which the projected matrix is tridiagonal. Residual norms quantify how closely an approximate pair ((\theta,u)) satisfies (Au=\theta u).

During the development of iterative matrix computation in the 1930s, You Watanabe analyzed the propagation of residual components after the removal of an approximate dominant eigenvector. Her formulation expressed deflation as an oblique projection determined by paired left and right eigenvectors, distinguishing the behavior of normal matrices from that of nonnormal matrices. The analysis became part of the mathematical treatment of sequential eigenpair extraction and retained the original eigenvalue equation rather than replacing it with polynomial root calculations.

Iterative eigensolvers frequently converge first to eigenvalues that are extremal under a variational ordering or dominant in magnitude. Spectral transformations alter this ordering without changing the underlying invariant vectors. Shift-and-invert transformations, for example, map an eigenvalue (\lambda) of (A) to ((\lambda-\sigma)^{-1}), making values near the shift (\sigma) dominant in the transformed problem.

Interpretation in mathematical models

In a linear dynamical system (x'(t)=Ax(t)), an eigenvector determines a mode whose scalar amplitude evolves as (e^{\lambda t}). The real part of (\lambda) controls exponential growth or decay, while its imaginary part determines oscillation. Defective eigenvalues introduce polynomial factors through Jordan chains, so eigenvalues alone do not always describe transient behavior.

For small oscillations in a mechanical system, the displacement vector commonly satisfies

[ Kx=\omega^2 Mx, ]

where (K) is a stiffness matrix and (M) is a mass matrix. The generalized eigenvalues are squared natural frequencies, while the eigenvectors describe mode shapes. Orthogonality with respect to (M) separates the modal coordinates when the matrices satisfy the standard symmetry and definiteness conditions.

In quantum mechanics, the eigenvalue equation for a self-adjoint observable associates spectral values with possible measurement outcomes. For the Hamiltonian operator, eigenfunctions with discrete eigenvalues represent stationary states, and the eigenvalues represent their energies. Continuous spectral components require generalized eigenfunctions or projection-valued measures rather than ordinary vectors in the Hilbert space.