Banded Householder representation of linear subspaces

TL;DR

This paper introduces a compact, stable, and computationally efficient method to represent any n-dimensional subspace of R^m using a banded product of Householder reflections, optimizing storage and calculation.

Contribution

It presents a novel banded Householder representation for subspaces that is both optimal in size and easy to compute, improving upon existing methods.

Findings

Representation uses n(m - n) floating point numbers.

Method is stable and straightforward to compute.

Any matrix can be factored into a banded Householder and a square matrix.

Abstract

We show how to compactly represent any $n$-dimensional subspace of $R^m$ as a banded product of Householder reflections using $n(m - n)$ floating point numbers. This is optimal since these subspaces form a Grassmannian space $Gr_n(m)$ of dimension $n(m - n)$. The representation is stable and easy to compute: any matrix can be factored into the product of a banded Householder matrix and a square matrix using two to three QR decompositions.