  To understand the solution of a linear, time-invariant differential-algebraic
equation, one must analyze a matrix pencil (A,E) with singular E. Even when
this pencil is stable (all its finite eigenvalues fall in the left-half plane),
the solution can exhibit transient growth before its inevitable decay. When the
equation results from the linearization of a nonlinear system, this transient
growth gives a mechanism that can promote nonlinear instability. One might hope
to enrich the conventional large-scale eigenvalue calculation used for linear
stability analysis to signal the potential for such transient growth. Toward
this end, we introduce a new definition of the pseudospectrum of a matrix
pencil, use it to bound transient growth, explain how to incorporate a
physically-relevant norm, and derive approximate pseudospectra using the
invariant subspace computed in conventional linear stability analysis. We apply
these tools to several canonical test problems in fluid mechanics, an important
source of differential-algebraic equations.
