Lipschitz stability of $\gamma$-FOCS and RC canonical Jordan bases of real $H$-selfadjoint matrices under small perturbations

TL;DR

This paper proves that both FOCS and classical real canonical Jordan bases of real H-selfadjoint matrices are Lipschitz stable under small perturbations, extending previous stability results to new bases with conjugate symmetry.

Contribution

It establishes Lipschitz stability for the newly introduced FOCS bases of real H-selfadjoint matrices, complementing existing results for classical Jordan bases.

Findings

FOCS bases are Lipschitz stable under small perturbations.

Classical real canonical Jordan bases are Lipschitz stable.

Extends stability results to bases with conjugate symmetry.

Abstract

In 2008 Bella, Olshevsky and Prasad proved that the flipped orthogonal (FO) Jordan bases of H-selfadjoint matrices are Lipschitz stable under small perturbations. In 2022, Dogruer, Minenkova and Olshevsky considered the real case, and proved that for real H-selfadjoint matrices there exist a more refined bases called FOCS bases. In addition to flipped orthogonality they also possess the conjugate symmetric (CS) property. In this paper we prove that these new FOCS bases are Lipschitz stable under small perturbations as well. We also establish the Lipschitz stability for the classical real canonical Jordan bases.