# ODE: How to measure stiffness if the Jacobian has zero eigenvalues?

Say you have a system of ODE's where the Jacobian has one zero eigenvalue; what does that tell you about the stiffness of the system?

This case doesn't seem to be discussed in the cases I have been able to find (Stiffness ratio or stiffness index).

The zero eigenvalue is not particularly harmful. Think of the linear system $$\dot x = A x, x(0) = x_0$$ then the components of $x(t)$ decay in proportion to the (magnitude of the) eigenvalues of $A$ (assuming the eigenvalues are all non-negative), whereas the components of $x(t)$ in direction of eigenvectors that belong to zero eigenvalues simply stay the same. In other words, these components simply don't factor at all in the determination whether a system is stiff or not.