A Basis-Kernel Representation of Orthogonal Matrices

Abstract

In this paper we introduce a new representation of orthogonal matrices. We show that any orthogonal matrix can be represented in the form $Q = I - Y S Y^T$, which we call the basis-kernel representation of $Q$. In particular, we point out that the kernel $S$ can be chosen to be triangular and that a familiar representation of an orthogonal matrix as a product of Householder matrices can be readily deduced from a basis-kernel representation with triangular kernel. We also show that there exists, in some sense, a minimal orthogonal transformation between two subspaces of same dimension, an important application of which is on block elimination problems. We explore how the basis $Y$ determines the subspaces that $Q$ acts on in a nontrivial fashion, and how $S$ determines the way $Q$ acts on this subspace. Especially, there is a canonical representation that explicitly shows that $Q$ partitions {\bf R}$^n$ into three invariant subspaces in which it acts as the identity, a reflector, and a rotator, respectively. We also present a generalized Cayley representation for arbitrary orthogonal matrices, which illuminates the degrees of freedom we have in choosing orthogonal matrices acting on a predetermined subspace.