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

TL;DR

This paper links invariant subspaces of non-normal matrices to transient effects in their exponential and powers, offering geometric insights and analytical results for 2D and Jordan subspaces.

Contribution

It introduces a relationship between invariant subspaces and transient effects, with analytical results for specific subspace types, enhancing understanding of transient phenomena.

Findings

Transient effects occur when eigenvector angles are small.

Invariant subspaces can predict transient growth.

Geometric interpretation of transient effects is provided.

Abstract

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.