Show that P(k, j)M(i) = [In − P(k, j)m(i)eT i ]P(k, j) for matrices M(i) in XXXXXXXXXXand XXXXXXXXXXwhen both k and j exceed i. For M(k) ∗ = In − m(k) ∗ eT k (recall (3.4.9)), show that...

Show that P(k, j)M(i) = [In − P(k, j)m(i)eT i ]P(k, j) for matrices M(i) in (3.4.2) and (3.4.3) when both k and j exceed i.

For M(k) ∗ = In − m(k) ∗ eT k (recall (3.4.9)), show that

                          M(n−1) P(n − 1, jn−1)··· M(2) P(2, j2)M(1) P(1, j1)

                         = M(n−1) ∗ ··· M(1) ∗ P(n − 1, jn−1)··· P(1, j1).