Year: 2016
Type: preprint
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.
Source: arXiv (Cornell University)
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2494934032