# Approximate eigenvalues

[SOLVED] Approximate eigenvalues

## Homework Statement

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

## Homework 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

## 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.