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Math Help - Eigenvalues of a Householder matrix

  1. #1
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    Eigenvalues of a Householder matrix

    We will prove in class that for all of the eigenvalues of an n n orthogonal matrix, |\lambda_i|^2 = 1, and in general will be complex. Consider here the eigenvalue problem, Hx = \lambdax, for a Householder matrix.

    (a) Starting with the fact that |\lambda_i| ^2 = 1, how do we know that all eigenvalues of H are either +1 or (−1)?

    (b) Beginning with Hx = \lambdax, show that there are only two possibilities, \lambda_i = −1, or v^T e_i = 0, 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 \lambda = 1. Using (*) above, explain why this is so.
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  2. #2
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    Re: Eigenvalues of a Householder matrix

    (a) All eigenvalues of A are $\pm 1$ whenever A is unitary and Hermetian. Prove the Hermetian part of this (they already took care of the unitary part).

    (b) not sure...

    (c) I saw the answer but was not able to figure out a key part of the proof. If we let $\mathbf{x} = \mathbf{v},$ we get $F\mathbf{v} = -\mathbf{v}.$ That means that exactly one of the eigenvalues is -1, but I am not sure why.
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