Description
Problem 1:
Determine one eigenvalue of the following matrix using Rayleigh Quotient iteration, starting with initial guess
π£
(0) = [0 1]
T
and initial eigenvalue estimate π
(0) = (π£
(0)
)
π
π΄π£
(0)
. Terminate iteration after 3 steps, i.e., after
you obtain
( 3 )
ο¬
. What is the approximate eigenvector π£
(3)
? What is the error of each π
(π)
?
π΄ = [
β6 2
2 β3
]
Problem 2:
Perform the first two iterations of the QR algorithm (i.e., compute π΄
(2)
and πΜ (2)
) for the following matrix. How
close are the diagonal elements of π΄
(2)
to the eigenvalues of π΄?
π΄ = [
3 β1 0
β1 2 β1
0 β1 3
]
Problem 3:
Reduce the following matrix to Hessenberg form using Householder reflector.
π΄ = [
3 β2 4 4
β2 1 9 β4
4 9 2 β4
4 β4 β4 2
]
Problem 4:
Let Q and R be the QR factors of a symmetric tridiagonal matrix H. Show that the product πΎ = π
π 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.)