Unstable modes in projection-based reduced-order models: How many can there be, and what do they tell you?

Projection methods provide an appealing way to construct reduced-order models of large-scale linear dynamical systems: they are intuitively motivated and fairly easy to compute. Unfortunately, the resulting reduced models need not inherit the stability of the original system. How many unstable modes can these reduced models have? This note investigates this question, using theory originally motivated by iterative methods for linear algebraic systems and eigenvalue problems, and illustrating the theory with a number of small examples. From these results follow rigorous upper bounds on the number of unstable modes in reduced models generated via orthogonal projection, for both continuous- and discrete-time systems. Can anything be learned from the unstable modes in reduced-order models? Several examples illustrate how such instability can helpfully signal transient growth in the original system.