Systems of differential equations
Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing:
Where , and . If all quadratic eigenvalues of are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as
Where are the quadratic eigenvalues, are the right quadratic eigenvectors, and is a parameter vector determined from the initial conditions on and .
Stability theory for linear systems can now be applied, as the behavior of a solution depends explicitly on the (quadratic) eigenvalues.