# Efficient Reduction of Compressed Unitary plus Low-rank Matrices to Hessenberg form

## TL;DR

This paper introduces efficient numerical methods for reducing a compressed unitary plus low-rank matrix to Hessenberg form, enabling faster eigenvalue computations by exploiting structured decompositions and bulge chasing techniques.

## Contribution

It develops a novel structured decomposition called LFR for such matrices and provides a fast reduction algorithm with $O(n^2 k)$ complexity, improving eigenvalue computation efficiency.

## Key findings

- Reduction cost is $O(n^2 k)$ arithmetic operations.
- LFR decomposition enables efficient Hessenberg reduction.
- Eigenvalues can be computed using a fast QR algorithm after reduction.

## Abstract

We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-rank matrix $A=G+U V^H$, where $G\in \mathbb C^{n\times n}$ is a unitary matrix represented in some compressed format using $O(nk)$ parameters and $U$ and $V$ are $n\times k$ matrices with $k< n$. At the core of these methods is a certain structured decomposition, referred to as a LFR decomposition, of $A$ as product of three possibly perturbed unitary $k$ Hessenberg matrices of size $n$. It is shown that in most interesting cases an initial LFR decomposition of $A$ can be computed very cheaply. Then we prove structural properties of LFR decompositions by giving conditions under which the LFR decomposition of $A$ implies its Hessenberg shape. Finally, we describe a bulge chasing scheme for converting the initial LFR decomposition of $A$ into the LFR decomposition of a Hessenberg matrix by means of unitary transformations. The reduction can be performed at the overall computational cost of $O(n^2 k)$ arithmetic operations using $O(nk)$ storage. The computed LFR decomposition of the Hessenberg reduction of $A$ can be processed by the fast QR algorithm presented in [8] in order to compute the eigenvalues of $A$ within the same costs.