a) Show that the householder matrix H=I-2ww* is unitary ( where ||w||=1).
b) Given a vector x in C^n (C: complex numbers) and an integer k with 1<k<n, derive a formula for a Householder matrix with the property that (Hx)_i = 0 for i> k. Be sure to choose the signs so that the formula is numerically stable.
For a), I think H is a unitary matrix if H H* = Identity matrix I. Now I guess, the thing is to just multiply H and H* and see if I get the identity matrix. Now the question is
(I-2ww*)*= (I-2w*w)? Or is it rather (I-2ww*)*= (I-2ww*) ?
For b)I don't know how to even start that one. Any help is much appreciated.