# Jacobi-Davidson method on low-rank matrix manifolds

## TL;DR

This paper extends the Jacobi-Davidson eigenvalue method to low-rank matrix manifolds, reducing computational complexity and storage needs while maintaining effectiveness, and introduces a low-rank Rayleigh quotient iteration.

## Contribution

It generalizes the Jacobi-Davidson method for low-rank matrices and develops a corresponding low-rank Rayleigh quotient iteration, offering efficiency improvements.

## Key findings

- Reduced complexity compared to traditional methods
- Lower storage requirements
- Effective eigenvalue computations on low-rank matrices

## Abstract

In this work we generalize the Jacobi-Davidson method to the case when eigenvector can be reshaped into a low-rank matrix. In this setting the proposed method inherits advantages of the original Jacobi-Davidson method, has lower complexity and requires less storage. We also introduce low-rank version of the Rayleigh quotient iteration which naturally arises in the Jacobi-Davidson method.