Revisiting the low-rank eigenvalue problem

Abstract

In this paper, we are interested in the eigenproblem on the large and low-rank matrix S=ABH, where A,B∈ℂn×r are of full column rank and r≪n. To the best of our knowledge, there are no results on the relations between the Jordan decomposition and the Schur decomposition of BHA and those of ABH. Some known results are only on characteristic polynomials, elementary divisors, and Jordan blocks of ABH, and are purely theoretical and are not easy to use for computational purposes. Based on the Jordan decomposition and the Schur decomposition of the small matrix BHA∈ℂr×r, we consider how to derive those of the large matrix AHB∈ℂn×n in this work. The construction methods proposed are not only theoretical but also practical. Numerical experiments show the effectiveness of our theoretical results.