Eigenvalues of a Householder matrix

We will prove in class that for all of the eigenvalues of an n × n orthogonal matrix, = 1, and in general will be complex. Consider here the eigenvalue problem, Hx = x, for a Householder matrix.

(a) Starting with the fact that = 1, how do we know that all eigenvalues of H are either +1 or (−1)?

(b) Beginning with Hx = x, show that there are only two possibilities, = −1, or , for all i = 1, 2, . . . n (*), where e_i is an eigenvector.

(c) In general, only one eigenvalue can be −1. That is, for and n × n Householder matrix, one eigenvalue is (−1), and there is (n − 1) algebraic and geometric multiplicity for = 1. Using (*) above, explain why this is so.