Banded Householder representation of linear subspaces

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 $G{r}_n(m)$ 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.

arXiv comment5 pages, 1 figure, submitted to Linear Algebra and its Applications