A note on constructive treatment of eigenvectors

TL;DR

This paper proposes an approximate, constructive approach to eigenvectors that addresses numerical instability issues in classical methods, especially useful when matrices lack certain spectral properties.

Contribution

It introduces a new constructive method for eigenvectors that manages computational uncertainty, improving stability over traditional algorithms.

Findings

Provides a constructive eigenvector approach that is more stable.

Addresses limitations of classical eigenvector computation in unstable matrices.

Offers a practical alternative for matrices with challenging spectral properties.

Abstract

The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or Jordan normal forms. Unfortunately, numerical algorithms computing eigenvectors are prone to errors. Due to uncomputability of eigenpairs, perturbation theory and various regularization techniques only help if the matrix at hand possesses certain properties such as the absence of non-zero singular values, or the presence of a distinguishable gap between the large and small singular values. Posing such a requirement might be restrictive in some practical applications. In this note, we propose an alternative treatment of eigenvectors which is approximate and constructive. In comparison to classical eigenvectors whose computation is often prone to numerical instability, a constructive treatment allows addressing the computational uncertainty in a controlled way.