cw3
March 2, 2021
算法作业代写 Submission Deadline Wed, 10 March 11am. Question 1:Let A ∈ Rm×n with m ≥ n with SVD A = U ΣV T . Show that the eigenvalues of the matrixare ···
1 Coursework 3 算法作业代写
Submission Deadline Wed, 10 March 11am. Question 1:
Let A ∈ Rm×n with m ≥ n with SVD A = U ΣV T . Show that the eigenvalues of the matrix
are given as σj and −σj with σj being the jth singular value of A, and associated eigenvectors
Question 2: 算法作业代写
Let x0 ∈ Rn. Denote by e1 the first unit vector. Let v˜ = α x0 2e1 − x0, where α = 1 if the first
element of x0 is negative, and α = −1 otherwise. Set v = v˜/ v˜ 2. Define the matrix
H = I − 2vvT .算法作业代写
H is called a Householder reflector. Show that Hx0 = α x0 2e1 and H = H−1. Question 3:
Describe how Householder reflectors can be used to implement a QR Decomposition. Be as precise as possible. Then implement the following function.
Question 4: 算法作业代写
What is the role of the parameter α? Why do you need it for numerical stability?
Question 5:
As we did in the lecture for the Gram-Schmidt algorithm, investigate the numerical orthogonality of the matrix Q obtained from Householder QR and comment on your observations. What happens if you always choose α = 1?
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