Math 1080: Homework #9 solution

Description

Problem 1:
Determine one eigenvalue of the following matrix using Rayleigh Quotient iteration, starting with initial guess
𝑣
(0) = [0 1]
T
and initial eigenvalue estimate 𝜆
(0) = (𝑣
(0)
)
𝑇
𝐴𝑣
(0)
. Terminate iteration after 3 steps, i.e., after
you obtain
( 3 )

. What is the approximate eigenvector 𝑣
(3)
? What is the error of each 𝜆
(𝑘)
?
𝐴 = [
−6 2
2 −3
]
Problem 2:
Perform the first two iterations of the QR algorithm (i.e., compute 𝐴
(2)
and 𝑄̃ (2)
) for the following matrix. How
close are the diagonal elements of 𝐴
(2)
to the eigenvalues of 𝐴?
𝐴 = [
3 −1 0
−1 2 −1
0 −1 3
]
Problem 3:
Reduce the following matrix to Hessenberg form using Householder reflector.
𝐴 = [
3 −2 4 4
−2 1 9 −4
4 9 2 −4
4 −4 −4 2
]
Problem 4:
Let Q and R be the QR factors of a symmetric tridiagonal matrix H. Show that the product 𝐾 = 𝑅𝑄 is again a
symmetric tridiagonal matrix.
(Hint: Prove the symmetry of K. Show that Q has Hessenberg form and that the product of an upper triangular
matrix and a Hessenberg matrix is again a Hessenberg matrix. Then use the symmetry of K.)