Testing Eigen Values: How to Check Accuracy Before Finding Eigen Vectors

In summary, the conversation discusses how to test the correctness of eigenvalues in a matrix. The main method suggested is to solve the characteristic equation and use the resulting eigenvalues and eigenvectors to check the properties of the matrix. This includes checking if the matrix multiplied by the eigenvector is parallel to the eigenvector, if the trace of the matrix is equal to the sum of the eigenvalues, and if the determinant of the matrix is equal to the product of the eigenvalues. Other possible methods include using a calculus-enabled program to check the answers.
  • #1
cragar
2,552
3

Homework Statement



This is probably a really simple question , but how do i test my eigen values to see if there right ,---------- (A-tI)x=0 where t is an eigen value , I know how to test if my eigen vectors are correct but how do i test to see if my eigen values are right , before i find my eigen vectors.
 
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  • #2
Well, if you solved your characteristic equation correct, then your eigenvalues should be correct. I don't know how you could check them before solving the eigenvalue problem and using the eigenvectors. Eigenvalues have some certain properties, but I don't see how they could help you determine if you're right.
 
  • #3
I think you're going to have to calculate the Eigenvectors and then check to see if they are both right.

Otherwise, you can check your answers using a calculus-enabled program like Maple.
 
  • #4
ok thanks , so when i find my eigen values and eigen vectors i can use
Ax=tx to see if they are both right .
 
  • #5
Not than you can, you have to use it. :)
 
  • #6
Well there's any number of things you can do. First of all you should multiply your matrix with your eigenvector and see if the vector you get is parallel to the eigenvector.

Then the trace of the matrix should be the sum of the eigenvalues, so you can check that also. Then the determinant of the matrix should be the product of the eigenvalues, check that if you have a reasonably small matrix.
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used in linear algebra to describe certain properties of a matrix. Eigenvalues represent the scaling factor of the eigenvectors when the matrix is multiplied by them.

2. Why is it important to test eigenvalues before finding eigenvectors?

Testing eigenvalues is important because it helps to ensure the accuracy of the eigenvectors. If the eigenvalues are incorrect, then the corresponding eigenvectors will also be incorrect.

3. How do you test the accuracy of eigenvalues?

To test the accuracy of eigenvalues, you can use the characteristic equation for eigenvalues, which is det(A - λI) = 0. This equation should hold true for all eigenvalues of the matrix A. You can also use numerical methods to approximate the eigenvalues and compare them to the theoretical values.

4. Can you check the accuracy of eigenvalues without finding the eigenvectors?

Yes, it is possible to check the accuracy of eigenvalues without finding the corresponding eigenvectors. This can be done by using the characteristic equation and comparing the results to the theoretical values.

5. What are some common methods for finding eigenvalues?

Some common methods for finding eigenvalues include the characteristic equation, power iteration, and the QR algorithm. Other methods such as Jacobi method, Lanczos algorithm, and the divide and conquer method can also be used.

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