= Damped quadratic eigenvalue problem
{title2=$(\lambda^2M+\lambda C+K)r=0$}
The <damped quadratic eigenvalue problem> finds exponential motions of a linear mechanical model with <mass matrix> $M$, <damping matrix> $C$ and <stiffness matrix> $K$. Substituting $u(t)=r e^{\lambda t}$ into $M\ddot u+C\dot u+Ku=0$ gives
$$
Q(\lambda)r=0,\qquad Q(\lambda)=\lambda^2M+\lambda C+K,\qquad r\ne0.
$$
The dependence on $\lambda$ is quadratic, so solving the undamped problem $Kr=\omega^2Mr$ alone does not find the damped decay rates and mode shapes.<The Decoupling of Damped Linear Systems in Oscillatory Free Vibration by Fai Ma, Ali Imam and Matthias Morzfeld>[§3.1, equation(14), quadratic eigenvalue formulation]{ref}
A useful linearization sets $z=(r,\lambda r)^T$ and rewrites the problem as
$$
\begin{pmatrix}0&I\\-K&-C\end{pmatrix}z
=\lambda\begin{pmatrix}I&0\\0&M\end{pmatrix}z.
$$
This <generalized eigenvalue problem> has twice the displacement dimension. With invertible $M$, it becomes an ordinary first-order state eigenproblem by multiplication by the block mass inverse. Singular $M$ requires more care and cannot be handled by silently using $M^{-1}$.
For an original one-coordinate example in consistent units, take $M=1$, $C=2$ and $K=5$. Then $\lambda^2+2\lambda+5=0$, giving $\lambda=-1\pm2i$. Real free motions have the form
$$
u(t)=e^{-t}(A\cos2t+B\sin2t).
$$
The decay rate is $1$ and damped oscillation frequency is $2$, while the undamped <natural frequency> is $\sqrt5$. Increasing $C$ to6 instead gives real roots $-1,-5$, showing that the quadratic problem also describes nonoscillatory decay.
For real symmetric positive-definite $M,K$ and positive-semidefinite $C$, an oscillatory eigenpair satisfies a useful energy check. Multiply by the conjugate transpose $r^*$ and write $m=r^*Mr>0$, $c=r^*Cr\ge0$, $k=r^*Kr>0$. The imaginary part of $m\lambda^2+c\lambda+k=0$ gives $\operatorname{Re}\lambda=-c/(2m)\le0$ when $\operatorname{Im}\lambda\ne0$. In a multi-coordinate system, complex eigenvectors can encode relative <phase differences> between coordinates; <nonclassical damping> generally prevents replacing them by the undamped real mode shapes.
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