# Banded Householder representation of linear subspaces

### Abstract

We show how to compactly represent any $n$-dimensional subspace of $\mathbb{R}^m$ as a banded product of Householder reflections using $n(m-n)$ floating point numbers. This is optimal since these subspaces Grassmannian space $\operatorname{Gr}_n(m)$ 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.

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