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Householder Reflector

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1. Determine the eigenvalues, determinant, and singular values of a Householder reflector. Give algebraic proofs for your conclusions.
2. Suppose Q E C^n, llqll2 = 1
Set P = I - qq^H.
(a) Find R(P)
(b) Firrd l/(P).
(c) Find the eigenvalues of P.
Prove your clairns.
3. Let A E C^(m*n)}, m (greater or equal to) n, with rank(A) = n. Prove that the reduced QR factorization
A = QR with the normalization r_jj > 0 is unique.
4. Suppose A E C^(n*n) is invertible. Let A = Q*R and A^H*A : U^H*U be the QR and Cholesky factorizations of A and AHA, respectively, with the normalizations r_jj,u_jj > 0. Prove that R = U.
5. Let A E C^(m*n). Use the SVD to prove the following:
(a) rank (A^H*A) = rank (A*A^H) = rank (A) = rank (A^H),
(b) A^H*A and A*A^H have the same nonzero eigenvalues,
(c) If the eigenvectors w1 and w2 of A^H*A are orthogonal, then Awi and Aw2 are orthogonal.

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Solution Preview

A Householder reflector is a linear transformation R of the form

Rv = v - 2n<v,n>

where n is a unit vector. Now construct an orthonormal basis for the vector space V of R by choosing n = e_1 as the first vector in the basis and choosing ...

Solution Summary

We determine the eigenvalues, determinant and singular values of a Householder reflector.