Determine the eigenvalues-determinant and singular values

Question

Question:

Householder Reflector

1. Determine the eigenvalues, determinant, and singular values of a Householder reflector. Give algebraic proofs for your conclusions.

2. Suppose q ∈Cⁿ, ||q||₂ = 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∈C^mxn, m≥n, with rank(A) = n. Prove that the reduced QR factorization

      A = Q^{^}R^{^} with the normalization r_jj > 0 is unique.

4. Suppose A∈C^nxn is invertible. Let A = QR and A^HA=U^HU be the QR and Cholesky factorizations of A and A^HA, respectively, with the normalizations r_jj,u_jj > 0. Prove that R = U.

5. Let A∈C^mxn. Use the SVD to prove the following:

(a) rank (A^HA) = rank (AA^H) = rank (A) = rank (A^H),
(b) A^HA and AA^H have the same nonzero eigenvalues,
(c) If the eigenvectors w₁ and w₂ of A^HA are orthogonal, then Aw₁ and Aw₂ are orthogonal.