  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.
