  It is well known that the matrix exponential of a non-normal matrix can
exhibit transient growth even when all eigenvalues of the matrix have negative
real part, and similarly for the powers of the matrix when all eigenvalues have
magnitude less than 1. Established conditions for the existence of these
transient effects depend on properties of the entire matrix, such as the Kreiss
constant, and can be laborious to use in practice. In this work we develop a
relationship between the invariant subspaces of the matrix and the existence of
transient effects in the matrix exponential or matrix powers. Analytical
results are obtained for two-dimensional invariant subspaces and Jordan
subspaces, with the former causing transient effects when the angle between the
subspace's constituent eigenvectors is sufficiently small. In addition to
providing a finer-grained understanding of transient effects in terms of
specific invariant subspaces, this analysis also enables geometric
interpretations for the transient effects.
