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

Matthew G. Reuter

Home/Archive/arXiv

arXiv · 1909.05931

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

Matthew G. Reuter

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.SP

Keep this discovery

BibTeX RIS

Matthew G. Reuter. 2019-09-12. Associating the Invariant Subspaces of a Non-Normal Matrix with Transient Effects in its Matrix Exponential or Matrix Powers. https://arxiv.org/abs/1909.05931

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Regular hyperbolic tilings have no $\ell^2$ eigenfunctions

We show that the adjacency operator of the $1$-skeleton of any regular tiling of the hyperbolic plane has no nonzero square-integrable eigenfunctions. As a consequence, the same holds for every infinite connected regular graph admitting a proper planar embedding with regular dual.

math.SP↗

arXiv2026

Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension

This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded $C^{1,1}$ domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive $q$-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are $q>d-2$ when the observation set is contained in a flat boundary patch and $q>d-1-c_{d+1}$ on a general $C^{1,1}$ boundary, where $c_{d+1}>0$ depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive $q$-dimensional Hausdorff content for some $q>d-1$, yielding logarithmic stability from full-time observations for general sources in $H_0^1(\Omega)$ and H\"older stability from terminal-time observations for sources in a suitable spectral Gevrey class.

math.SP↗

arXiv2026

Resolvent bounds and eigenvalue estimates of generalized Schr\"odinger operators with complex potentials on compact manifolds

We extend Cuenin's compact-manifold spectral bounds for Schr\"odinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators $P+V$, where $P$ is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and $V$ is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for $P$ imply $L^p$-$L^{p'}$ resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.

math.SP↗