Matching spectral lower bound for reflector budgets

Establish a matching spectral lower bound for the number of Householder reflectors required to realize the relevant orthogonal transformation under the rank condition supplied by a reflector budget.

Background

The paper constructs partial orthogonal preconditioners using at most twice the target subspace dimension in Householder reflectors and notes that any product of r reflectors satisfies rank(U-I)≤r. This rank condition is only necessary; the unresolved issue is whether additional geometric information can yield a matching spectral lower bound on the reflector budget.