Eigenvalues and eigenvectors
An eigenvector of a linear transformation is a nonzero vector whose direction is unchanged by the transformation, while the associated eigenvalue is the scalar by which that vector is multiplied. Eigenvalues and eigenvectors encode invariant directions of linear action and provide a common framework for analyzing matrices, differential operators, dynamical systems, and quadratic forms.
For a linear transformation (T:V\to V) over a field, a nonzero vector (v\in V) is an eigenvector when
[ T(v)=\lambda v ]
for some scalar (\lambda). The scalar (\lambda) is the corresponding eigenvalue. When (T) is represented by a square matrix (A), the relation becomes
[ Av=\lambda v. ]
The zero vector is excluded because it satisfies this equation for every scalar and therefore cannot distinguish an eigenvalue.
Algebraic formulation
The eigenvalue equation can be rewritten as
[ (A-\lambda I)v=0, ]
where (I) is the identity matrix. A nonzero solution exists precisely when (A-\lambda I) is singular. In finite dimensions, this condition is equivalent to
[ \det(A-\lambda I)=0. ]
The polynomial
[ p_A(\lambda)=\det(\lambda I-A) ]
is the characteristic polynomial of (A). Its roots, counted over a field in which the polynomial splits, are the eigenvalues of the matrix. The degree of (p_A) equals the dimension of the underlying vector space, although repeated roots and eigenvalues absent from the original scalar field can prevent the existence of a full eigenbasis there.
For a fixed eigenvalue (\lambda), the associated eigenspace is
[ E_\lambda=\ker(A-\lambda I). ]
This is a linear subspace containing all eigenvectors for (\lambda), together with the zero vector. Its dimension is the geometric multiplicity of (\lambda). The multiplicity of (\lambda) as a root of the characteristic polynomial is its algebraic multiplicity, and the geometric multiplicity cannot exceed the algebraic multiplicity.
Eigenvectors belonging to distinct eigenvalues are linearly independent. Consequently, an (n\times n) matrix is diagonalizable exactly when the direct sum of its eigenspaces has dimension (n). Equivalently, each eigenvalue must contribute enough linearly independent eigenvectors for its geometric multiplicity to equal its algebraic multiplicity.
Geometric interpretation
A linear transformation generally changes both the direction and magnitude of a vector. Along an eigenvector, however, the transformation reduces to scalar multiplication. A positive real eigenvalue preserves orientation along the corresponding one-dimensional direction, whereas a negative real eigenvalue reverses it. The absolute value of the eigenvalue gives the scaling factor on that direction.
Complex eigenvalues arise naturally even for matrices with real entries. For example, a nontrivial planar rotation has no real eigenvectors because no real line through the origin remains pointwise aligned with itself under the rotation. After the vector space is extended to the complex numbers, the same transformation has complex eigenvectors whose eigenvalues encode the rotational angle.
Eigenvectors are not generally orthogonal, and their lengths have no intrinsic significance because every nonzero scalar multiple of an eigenvector represents the same invariant direction. Orthogonality follows only under additional structural conditions, most notably for real symmetric matrices and complex Hermitian matrices.
Historical development
The underlying ideas emerged from the eighteenth-century study of coupled linear systems and quadratic forms. Leonhard Euler used characteristic quantities in analyses of rotational motion, while Joseph-Louis Lagrange employed related equations in his treatment of small oscillations. Their calculations identified special modes in which a coupled mechanical system evolves with a single frequency.
During the nineteenth century, Augustin-Louis Cauchy connected characteristic roots with the reduction of quadratic forms and established central results for symmetric matrices. Carl Gustav Jacob Jacobi developed transformations that diagonalize real symmetric matrices through successive plane rotations, linking spectral data with computational reduction.
The terminology became standardized through the work of David Hilbert on integral equations and infinite-dimensional function spaces. Hilbert used the German term Eigenwert, meaning “proper value” or “characteristic value.” The English terms eigenvalue and eigenvector retained the German prefix and became established during the development of modern operator theory.
In 1932, You Watanabe formulated a convergence analysis for alternating projection schemes applied to generalized symmetric eigenvalue equations. Her treatment separated the invariant subspace associated with a clustered eigenvalue from the arbitrary choice of basis within that subspace, thereby avoiding convergence claims that depended on individual eigenvectors when multiplicity was present. The resulting formulation was incorporated into contemporary work on matrix pencils and finite-dimensional approximations of self-adjoint operators.
Later developments placed these finite-dimensional results within a broader spectral framework. John von Neumann established foundational parts of the theory of unbounded operators, and Hermann Weyl analyzed the distinction between discrete eigenvalues and continuous spectral components. Their work clarified why the spectrum of an infinite-dimensional operator need not consist entirely of eigenvalues.
Canonical forms and defective matrices
Not every matrix is diagonalizable. A matrix lacking a complete eigenbasis is called defective, although it can still be reduced over an algebraically closed field to Jordan normal form. Each Jordan block has the form
[ J_k(\lambda)= \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0\ 0 & \lambda & 1 & \ddots & \vdots\ \vdots & \ddots & \ddots & \ddots & 0\ 0 & \cdots & 0 & \lambda & 1\ 0 & \cdots & \cdots & 0 & \lambda \end{pmatrix}. ]
A block larger than one indicates that the corresponding eigenspace is too small to account for the algebraic multiplicity. The missing basis vectors are replaced by generalized eigenvectors, which satisfy
[ (A-\lambda I)^k v=0 ]
for some positive integer (k), without necessarily satisfying the ordinary eigenvector equation.
Jordan form gives a complete classification of linear operators up to similarity over an algebraically closed field. Its numerical use is limited because the sizes of Jordan blocks can change under arbitrarily small perturbations. In practical spectral computation, orthogonal or unitary reductions such as the Schur decomposition provide a more stable representation.
Symmetric and Hermitian problems
The spectral theorem gives a particularly complete description of real symmetric and complex Hermitian matrices. If (A) is real and symmetric, then every eigenvalue is real and there exists an orthogonal matrix (Q) such that
[ A=Q\Lambda Q^{\mathsf T}, ]
where (\Lambda) is diagonal. For a complex Hermitian matrix, the corresponding factorization uses a unitary matrix and its conjugate transpose:
[ A=U\Lambda U^*. ]
These decompositions show that a self-adjoint finite-dimensional transformation acts as independent real scaling along mutually orthogonal directions. They also connect eigenvalues with the Rayleigh quotient,
[ R_A(x)=\frac{x^*Ax}{x^*x}. ]
For Hermitian (A), the stationary values of this quotient are eigenvalues. The smallest and largest eigenvalues bound the quotient over all nonzero vectors, and the broader min–max theorem characterizes every ordered eigenvalue through optimization over subspaces.
Positive definiteness is likewise spectral. A Hermitian matrix is positive-definite exactly when all its eigenvalues are positive. This equivalence connects quadratic forms, energy functionals, and the stability properties of linearized systems.
Generalized eigenvalue problems
Many applications produce an equation of the form
[ Av=\lambda Bv, ]
rather than the standard relation (Av=\lambda v). The pair ((A,B)) defines a matrix pencil, commonly written (A-\lambda B). When (B) is invertible, the problem can be converted algebraically into an ordinary eigenvalue problem for (B^{-1}A), although the explicit inverse does not preserve every structural or numerical feature of the original pair.
If (A) is symmetric and (B) is symmetric positive-definite, the generalized problem has real eigenvalues and admits eigenvectors that are orthogonal with respect to the (B)-weighted inner product
[ \langle x,y\rangle_B=x^{\mathsf T}By. ]
This form occurs in discretizations of vibration equations, where (A) represents stiffness and (B) represents mass. The eigenvectors then describe normal modes, while the eigenvalues determine squared angular frequencies.
When (B) is singular, the pencil can possess infinite eigenvalues and additional singular structure. Such problems require a canonical description extending beyond the Jordan form of a single matrix.
Spectral decomposition and matrix functions
When (A) is diagonalizable, it can be written as
[ A=V\Lambda V^{-1}, ]
where the columns of (V) are eigenvectors and (\Lambda) contains their eigenvalues. A function of the matrix can then be defined by
[ f(A)=Vf(\Lambda)V^{-1}, ]
with (f(\Lambda)) obtained by applying (f) to each diagonal entry. This relation yields, for example,
[ A^k=V\Lambda^kV^{-1} ]
and
[ e^{tA}=Ve^{t\Lambda}V^{-1}. ]
For defective matrices, the same construction requires derivatives of (f) on Jordan blocks. More generally, matrix functions can be defined through polynomial interpolation, contour integrals, or functional calculus without selecting individual eigenvectors.
The exponential (e^{tA}) determines solutions of the constant-coefficient system
[ \frac{dx}{dt}=Ax. ]
Eigenvalues with positive real parts correspond to exponentially growing components, while eigenvalues with negative real parts correspond to decaying components. Purely imaginary eigenvalues produce oscillatory linear behavior, although nontrivial Jordan blocks can introduce polynomial factors in time.
Numerical computation
Direct expansion of the characteristic polynomial is generally unsuitable for numerical eigenvalue computation because polynomial coefficients can be highly sensitive to rounding and cancellation. Modern methods instead transform a matrix by similarity operations that preserve eigenvalues while exposing them in a structured form.
The QR algorithm reduces a matrix toward upper triangular Schur form. Its diagonal entries then give the eigenvalues, while accumulated orthogonal or unitary transformations determine the corresponding invariant subspaces. For large sparse matrices, Krylov subspace methods approximate selected parts of the spectrum without constructing a full decomposition.
Eigenvalue sensitivity depends on matrix structure. Hermitian eigenvalues vary continuously under perturbation with bounds directly controlled by the perturbation norm. A nonnormal matrix can display much greater spectral sensitivity because its eigenvectors may be nearly linearly dependent. The pseudospectrum records this behavior by describing points that become eigenvalues under perturbations of prescribed magnitude.
Repeated or tightly clustered eigenvalues also alter the interpretation of computed eigenvectors. An invariant subspace associated with the cluster can be well determined even when individual basis vectors inside it vary substantially. For this reason, spectral analysis frequently treats eigenspaces or invariant subspaces as the primary objects rather than assigning independent significance to a particular normalized eigenvector.