(a) Show that if u ∊ Rn is a unit vector, then the n × n matrix Q = 1 - 2uuT is an orthogonal matrix, known as an elementary reflection or Householder matrix.
(b) Write down the elementary reflection matrices corresponding to the following unit vectors:
(i) (1, 0)T
(ii) (3/5, 4/5)T
(iii) (0, 1, 0)T
(iv) (1/√2, 0, - 1/√2)T
(c) Prove that
(i) Q is symmetric
(ii) Q-1 = Q
(iii) Qu = - u
(d) Find all vectors fixed by an elementary reflection matrix, i.e., Q v = v-first for the matrices in part (b), and then in general.