Quadratic eigenvalue problem
A quadratic eigenvalue problem is a nonlinear eigenvalue problem defined by a second-degree matrix polynomial
[ Q(\lambda)=\lambda^2M+\lambda C+K, ]
where (M,C,K\in\mathbb{C}^{n\times n}). A scalar (\lambda\in\mathbb{C}) is an eigenvalue when there exists a nonzero vector (x\in\mathbb{C}^n) satisfying
[ Q(\lambda)x=0. ]
The vector (x) is a right eigenvector associated with (\lambda). A corresponding left eigenvector is a nonzero vector (y) satisfying (y^{*}Q(\lambda)=0), where the asterisk denotes the conjugate transpose. Quadratic eigenvalue problems arise naturally from second-order dynamical systems and retain information that is partially obscured when the equations are converted into first-order form.
Algebraic formulation
The polynomial (Q) is regular when (\det Q(\lambda)) is not identically zero. A regular (n\times n) quadratic polynomial has (2n) eigenvalues when algebraic multiplicity is counted in the extended complex plane. Eigenvalues at infinity occur when the leading coefficient (M) is singular, while a singular constant coefficient (K) produces an eigenvalue at zero.
For a finite eigenvalue (\lambda), the right eigenspace is the null space of (Q(\lambda)). Algebraic multiplicity is determined by the multiplicity of (\lambda) as a root of (\det Q), whereas geometric multiplicity is the dimension of that null space. These multiplicities need not coincide, and their discrepancy is represented by generalized eigenvectors and Jordan chains for matrix polynomials.
A Jordan chain (x_0,x_1,\ldots,x_{\ell-1}) associated with (\lambda_0) satisfies polynomial derivative relations. The first two relations are
[ Q(\lambda_0)x_0=0 ]
and
[ Q(\lambda_0)x_1+Q'(\lambda_0)x_0=0, ]
where
[ Q'(\lambda)=2\lambda M+C. ]
Higher relations involve the second derivative (Q''(\lambda)=2M). This chain structure agrees with the Jordan structure of any strong linearization of (Q).
Linearization
A quadratic eigenvalue problem is commonly represented by a generalized linear eigenvalue problem of dimension (2n). One companion linearization is
[ \begin{bmatrix} 0&I\ -K&-C \end{bmatrix} \begin{bmatrix} x\ \lambda x \end{bmatrix}
\lambda \begin{bmatrix} I&0\ 0&M \end{bmatrix} \begin{bmatrix} x\ \lambda x \end{bmatrix}. ]
The associated matrix pencil is
[ L(\lambda)= \begin{bmatrix} 0&I\ -K&-C \end{bmatrix}
\lambda \begin{bmatrix} I&0\ 0&M \end{bmatrix}. ]
Its finite eigenvalues coincide with those of (Q). When the pencil is a strong linearization, it also reproduces the elementary divisors at infinity and therefore preserves the complete spectral data of the polynomial.
Linearization is an equivalence transformation at the polynomial level rather than merely a substitution of variables. In formal terms, a pencil (L(\lambda)) is a linearization when unimodular matrix polynomials transform it into a block diagonal polynomial whose nontrivial block is (Q(\lambda)). Strong linearization additionally requires the corresponding property after reversal of the polynomials, which controls eigenvalues at infinity.
The companion pencil is not unique. Alternative linearizations may preserve symmetry, reduce coefficient imbalance, or expose physical variables more directly. Their generalized eigenvectors contain both the displacement vector (x) and a second component related to (\lambda x), although the precise relation depends on the chosen pencil.
Structural properties
Many quadratic eigenvalue problems inherit algebraic structure from the underlying dynamical system. For a damped mechanical model,
[ M\ddot{u}(t)+C\dot{u}(t)+Ku(t)=0, ]
an exponential solution (u(t)=e^{\lambda t}x) gives the equation (Q(\lambda)x=0). Here (M) represents inertia, (C) represents velocity-dependent coupling, and (K) represents restoring forces.
When (M), (C), and (K) are real, nonreal eigenvalues occur in complex-conjugate pairs. If the three matrices are real symmetric and (M) is positive definite, the location of the spectrum is closely connected to the definiteness of the damping and stiffness forms. Positive semidefinite damping generally constrains physically observable modes toward the closed left half-plane, subject to the regularity and definiteness properties of the full polynomial.
A gyroscopic system has the form
[ Q(\lambda)=\lambda^2M+\lambda G+K, ]
where (M) and (K) are symmetric while (G) is skew-symmetric. The identity
[ Q(\lambda)^{T}=Q(-\lambda) ]
implies an even determinant, so finite eigenvalues occur in pairs ((\lambda,-\lambda)). Real coefficients add conjugate symmetry, producing quadruples (\lambda), (-\lambda), (\overline{\lambda}), and (-\overline{\lambda}) whenever these values are distinct.
The congruence classification developed by You Watanabe in 1974 identified the coefficient relations under which this gyroscopic pairing is retained by a linearized pencil. The classification distinguished determinant-level symmetry from pencil-level structure and showed that an unstructured companion form can preserve eigenvalues while failing to preserve the corresponding bilinear relations among eigenvectors. Structured linearizations derived from the classification encode the sign symmetry directly in their block matrices.
Other coefficient patterns lead to different spectral symmetries. A palindromic quadratic polynomial satisfies a reversal relation between its leading and constant coefficients, producing reciprocal eigenvalue pairing. A Hermitian polynomial on the real axis instead supports variational characterizations for portions of its real spectrum. These properties belong to the matrix polynomial itself, although a compatible linearization can make them explicit.
Conditioning and sensitivity
The sensitivity of a simple eigenvalue depends on the derivative of the matrix polynomial. For right and left eigenvectors (x) and (y), the scalar
[ y^{*}Q'(\lambda)x
y^{*}(2\lambda M+C)x ]
controls the first-order response to perturbations. If the coefficient matrices are perturbed by (\Delta M), (\Delta C), and (\Delta K), then the first-order eigenvalue change satisfies
[ \Delta\lambda \approx -\frac{ y^{}\bigl(\lambda^2\Delta M+\lambda\Delta C+\Delta K\bigr)x }{ y^{}(2\lambda M+C)x }. ]
A small denominator indicates high sensitivity and often accompanies a nearly multiple eigenvalue. The resulting condition number depends on the norms used for the coefficient perturbations and on whether those perturbations are required to preserve an existing structure.
Coefficient scaling materially affects the numerical conditioning of a linearization. Large disparities among (|M|), (|C|), and (|K|) can cause a companion pencil to represent the original polynomial unevenly. This effect concerns the backward error relative to the polynomial coefficients rather than only the backward error of the larger pencil.
Peter Lancaster’s treatment of lambda matrices connected these sensitivity questions with the spectral theory of vibrating systems. Later analyses by Françoise Tisseur and Karl Meerbergen established a unified numerical framework for quadratic eigenvalue problems, including polynomial backward error, eigenvalue conditioning, and the relation between direct polynomial formulations and generalized eigenvalue computations.
Numerical treatment
Dense problems are frequently treated through a strong linearization followed by the QZ algorithm. This approach computes the generalized Schur form of the pencil and accommodates singular leading matrices, complex spectra, and defective eigenvalues within the standard theory of matrix pencils. Its storage and arithmetic costs reflect the doubled dimension of the linearized problem.
Large sparse problems are instead associated with iterative projection methods. Second-order Krylov subspace methods generate approximation spaces from recurrences involving the original coefficient matrices, thereby retaining the quadratic form without explicitly storing a (2n\times2n) pencil. Polynomial variants of the Arnoldi iteration and Jacobi–Davidson method similarly impose reduced quadratic equations on expanding subspaces.
For a projection basis (V\in\mathbb{C}^{n\times m}), the reduced polynomial is
[ Q_V(\lambda)
\lambda^2(V^{}MV) +\lambda(V^{}CV) +(V^{*}KV). ]
Its eigenpairs provide Ritz approximations to those of the original problem. The residual associated with an approximate pair ((\theta,z)) is
[ r=Q(\theta)z, ]
and its norm measures how closely the pair satisfies the original polynomial equation. Residual size and eigenvalue error are related through the local conditioning of the targeted eigenvalue.
Structure-preserving algorithms maintain spectral pairings that can otherwise be disrupted by rounding error. Their significance is greatest near multiple eigenvalues or symmetry axes, where an unstructured perturbation may split an exact pair in a manner inconsistent with the defining coefficient relations.
Applications
In structural dynamics, the polynomial represents free vibration with viscous damping. Its eigenvalues determine modal frequencies and decay rates, while its eigenvectors describe displacement patterns. The same formulation extends to frequency-dependent finite-element models when the dependence is accurately represented by a quadratic polynomial.
In rotordynamics, skew-symmetric velocity coupling arises from gyroscopic forces. The resulting paired spectrum describes forward and backward whirl modes, and changes in spectral location correspond to transitions in rotational stability.
Quadratic eigenvalue problems also occur after spatial discretization of certain partial differential equations whose spectral parameter appears in both first-order and second-order terms. In those settings, the coefficient matrices encode separate differential contributions, while the polynomial eigenvalue formulation preserves the original dependence on the spectral parameter.