How can I accurately find eigenvalues for a Jordan canonical form matrix?

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In summary, the conversation is about finding the Jordan canonical form for a given matrix and having difficulties with finding the eigenvalues. The person tried a method using the determinant but got a different answer compared to an online calculator. They were advised to use the other method and make sure to change the signs of the non-diagonal entries. The conversation also mentions that the topic of Jordan canonical form is more suitable for the "Calculus and Beyond" section.
  • #1
pinguhash
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Homework Statement


Ok I was working with finding Jordan canonical form...
Here is the matrix I was working on:
| 1 1 1 |
|-1 -1 -1 |
| 1 1 0 |

I am having problem with finding eigenvalues... below is the attempt to solution
I was not getting the right answer. So, when I used online calculator to find the eigenvalue it was comletely different from what I got!

2. The attempt at a solution

|λ-1 , 1 , 1 |
|-1 , λ+1 , -1 |
|1 , 1 , λ |

So, I got values something λ3 = 4

The values from online calculator was λ3 = 0

Please help me in finding how they got eigenvalues all 0.
 
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  • #2
You can either do det(A - xI) = 0, or det(xI - A) = 0, where A is your matrix and x I used instead of lambda. You chose to go with the second one. but forgot to change the signs of the non-diagonal entries.

Btw, I think this should go into "Calculus and Beyond". JCF certainly isn't precalculus material :P
 
  • #3
oh sorry for wrong section, I am new to this forum...
 

Related to How can I accurately find eigenvalues for a Jordan canonical form matrix?

1. What are eigenvalues and why are they important in JCF?

Eigenvalues are a set of numbers that represent the scaling factor of a vector when it is transformed by a matrix. They are important in JCF (Jordan Canonical Form) because they provide a way to simplify and understand the structure of a matrix.

2. How do you find eigenvalues for JCF?

The process of finding eigenvalues for JCF involves finding the roots of the characteristic polynomial of the matrix. This polynomial is obtained by subtracting the eigenvalue from the diagonal elements of the matrix and setting the resulting determinant equal to 0. The resulting eigenvalues will be the roots of the polynomial.

3. Can you provide an example of finding eigenvalues for JCF?

Sure, consider the matrix A = [2 1; 1 2]. The characteristic polynomial would be det(A - λI) = (2-λ)(2-λ) - 1 = 4 - 4λ + λ^2 - 1 = λ^2 - 4λ + 3. Setting this equal to 0 and solving for λ, we get eigenvalues of 3 and 1.

4. What is the significance of the Jordan Canonical Form in linear algebra?

The Jordan Canonical Form is significant in linear algebra because it allows us to understand the structure of a matrix and its corresponding linear transformation. It also provides a way to simplify calculations and solve problems involving matrices.

5. Are there any limitations or challenges in finding eigenvalues for JCF?

There are a few limitations and challenges in finding eigenvalues for JCF. One limitation is that not all matrices have a JCF, and some may have a more complex form. Another challenge is that finding the roots of higher order polynomials can be difficult and may require the use of numerical methods.

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