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It is well known that the matrix exponential of a non-normal matrix canexhibit transient growth even when all eigenvalues of the matrix have negativereal part, and similarly for the powers of the matrix when all eigenvalues havemagnitude less than 1. Established conditions for the existence of thesetransient effects depend on properties of the entire matrix, such as the Kreissconstant, and can be laborious to use in practice. In this work we develop arelationship between the invariant subspaces of the matrix and the existence oftransient effects in the matrix exponential or matrix powers. Analyticalresults are obtained for two-dimensional invariant subspaces and Jordansubspaces, with the former causing transient effects when the angle between thesubspace's constituent eigenvectors is sufficiently small. In addition toproviding a finer-grained understanding of transient effects in terms ofspecific invariant subspaces, this analysis also enables geometricinterpretations for the transient effects.

Associating the Invariant Subspaces of a Non-Normal Matrix with Transient Effects in its Matrix Exponential or Matrix Powers