The subspaces spanned by Householder vectors associated with an orthogonal or a symplectic matrix

The Cartan-Dieudonné-Scherk Theorem guarantees that every complex orthogonal matrix can be written as a product of matrices of the form HS,u≡I−uuTS, where S=I and u∈Cn satisfies uTu=2; moreover, every complex symplectic matrix can be written as a product of matrices of the form HS,u≡I−uuTS where S=J=[0I−I0] and u≠0. Let a nonempty V⊆Cn be given. The S-orthogonal complement of V is VS={z∈Cn|wTSz=0 for all w∈V}. The image of an n-by-n complex matrix A is the set of all z∈Cn for which there is an x∈Cn such that z=Ax and is denoted by Im(A). Let S=I or S=J. Suppose that Q=HS,u1HS,u2⋯HS,ur. Set U=span{u1,u2,…,ur}. We study the relationship between Q, U, and Im(Q−I). Suppose that r is minimal. We show that if dim⁡(U)=r, then Im(Q−I)=U. We also show that S(Q−I) is not skew symmetric if and only if dim⁡(U)=r. Let W=Im(Q−I). We show that a relationship between W and WS determines the Jordan structure of Q, in particular, we show that (Q−I)2=0 if and only if W⊆WS.