  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.
