12.1. For A defined as in (12.5), smoothing matrices Sk, and n-vectors k for k = 1,...,p,...

12.1. For A defined as in (12.5), smoothing matrices Sk, and n-vectors γk for k = 1,...,p, let Ik be the space spanned by vectors that pass through Sk unchanged (i.e., vectors v satisfying Skv = v). Prove that Aγ = 0 (where γ = (γ1 γ2 ... γp)

T) if and only if

γk ∈ Ik for all k and p k=1 γk = 0.