Approximate Eigenvalues for [4 3] [3 5]

In summary, the conversation was about using the QR method to approximate the eigenvalues of a given matrix and comparing them to the actual values. Two methods were attempted, but both resulted in a non-symmetric matrix at the second iteration. Ultimately, the problem was solved by the person and they no longer needed assistance.
  • #1
angelz429
24
0
[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.
 
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  • #2
Well, since no one can help... :(

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

1. What are approximate eigenvalues?

Approximate eigenvalues are values that are close to the actual or exact eigenvalues of a given matrix. They are obtained through various numerical methods and are used to approximate the behavior of real-world systems.

2. Why are approximate eigenvalues important?

Approximate eigenvalues are important because they provide a way to analyze the behavior of a system without needing to solve for the exact eigenvalues, which can be computationally expensive. They are also useful for estimating the stability and dynamics of a system.

3. How are approximate eigenvalues calculated?

There are several numerical methods for calculating approximate eigenvalues, such as the power method, inverse iteration, and the Jacobi method. These methods involve iteratively solving equations and approximating the eigenvalues based on the results.

4. What factors can affect the accuracy of approximate eigenvalues?

The accuracy of approximate eigenvalues can be affected by the choice of numerical method, the size and complexity of the matrix, and the precision of the computing system. Convergence criteria and round-off errors can also impact the accuracy.

5. How are approximate eigenvalues used in real-world applications?

Approximate eigenvalues are used in a variety of real-world applications, such as in engineering, physics, and economics. They can be used to analyze the stability of structures, predict the behavior of quantum systems, and model economic systems. They are also used in machine learning and data analysis to reduce the dimensionality of data and identify important features.

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