[SOLVED] Approximate eigenvalues

angelz429

1. The problem statement, all variables and given/known data

Use some QR method to approximate the eigenvalues of
[4 3]
[3 5]
and compare with the actual values.

The actual values are (9±√37)/2


2. Relevant equations

A(0)=Q(0)R(0)
A(1)=R(0)Q(0)

A-α(0)I=Q(0)R(0)
A(1)=R(0)Q(0) + α(0)I


3. The attempt at a solution

A(0) = [4 3] = [(4/5) (-3/5)] [5 (27/5)]
[3 5] [(3/5) (4/5)] [0 (11/5)]

A(1)= [5 (27/5)] [(4/5) (-3/5)] = [(181/25) (33/25)]
[0 (11/5)] [(3/5) (4/5)] [(33/25) (44/25)]

When I get to A(2), its no longer symmetric

Same if I try it the Shifted QR Method

A(0) = [4 3] α(0) = 5
[3 5]

A(0)-α(0)I = [-1 3] = [(-1/sqrt 10) (3/sqrt 10)] [(sqrt 10) (-3/sqrt 10)]
[3 0] [(3/sqrt 10) (1/sqrt 10)] [ 0 (9/sqrt 10)]

A(1) = [(sqrt 10) (-3/sqrt 10)] [(-1/sqrt 10) (3/sqrt 10)] + [5 0]
[ 0 (9/sqrt 10)] [(3/sqrt 10) (1/sqrt 10)] [0 5]

= [(51/10) (27/10)] α(1) = (59/10)
[(27/10) (59/10)]

When I get to A(2), its no longer symmetric

So I'm not sure what I'm doing wrong... or if there's another QR method to solve it.

angelz429

Well, since no one can help.... :(

I don't need it anymore, thanks for looking!