A Basis-Kernel Representation of Orthogonal Matrices

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 SY T , which we call the basis-kernel representation of Q. We show that the kernel S can be chosen to be triangular and show how the familiar representation of an orthogonal matrix as a product of Householder matrices can be directly derived from a representation with triangular kernel. We also show that there exists an, in some sense, minimal orthogonal transformation for solving the block elimination problem. 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. We derive a canonical representation that explicitly shows how Q partitions R n into three invariant subspaces in which it acts as the identity, a reeector, and a rotator, respectively. We also derive 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.