Canonical form and change of coordinates for a matrix

AI Thread Summary
The discussion revolves around finding the canonical forms and change of coordinates for a specific 6x6 matrix in linear algebra. The user is struggling with the process due to a lack of resources and guidance from their professor, who has only covered 2x2 matrices. They have calculated the determinant and identified the roots, including real and complex values, but are uncertain about how to proceed with eigenvector evaluation and the application of canonical forms for the larger matrix. The conversation highlights the need for clarity on handling complex roots and integrating them into the canonical form alongside the real roots. Overall, the user seeks assistance in navigating these advanced linear algebra concepts.
jejaques
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Hello! I'm trying to do some linear algebra. I have an insane Russian teach whose English is, uh, lacking.. so I'd appreciate any help with these I can get here!

Homework Statement


Find the canonical forms for the following linear operators and the matrices for the corresponsing change of coordinates.

Here is the 6x6 matrix:
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
-1 0 0 -2 0 0


Homework Equations





The Attempt at a Solution


I know I have to do subtract \lambda on the diagonal, take the determinant, find the roots by solving for the \lambda values, and then plug them in one at a time to find the different \zeta, turn that into a change of coordinates, and then depending on case, put it into canonical form...

Unfortunately, my professor has only shown us the various \lambda cases for 2 x 2 matrices and because we can "look everything up on google," we have no book!

A couple questions: Can I simplify this or maybe turn it into the Jordan block? Can anyone point me to a similar problem, even? I've been searching for two hours, have searched through three free linear algebra e-books and am still lost =(

Thanks so much!
 
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Welcome to PF!

Hi jejaques! Welcome to PF! :smile:

(have a lambda: λ :wink:)
jejaques said:
Unfortunately, my professor has only shown us the various \lambda cases for 2 x 2 matrices and because we can "look everything up on google," we have no book!

Just put -λ down the diagonal, and calculate the determinant :smile:
 


tiny-tim said:
Hi jejaques! Welcome to PF! :smile:

(have a lambda: λ :wink:)


Just put -λ down the diagonal, and calculate the determinant :smile:


tiny-tim said:
Hi jejaques! Welcome to PF! :smile:

(have a lambda: λ :wink:)


Just put -λ down the diagonal, and calculate the determinant :smile:


Hello, and thanks for the welcome...

Yeah, my reasoning was in my "attempt at a solution" section. I subtracted \lambda from the diagonal and did the determinant; I just thought it was too much tedious stuff to post here, as I'm having problems further on.

The determinant is \lambda6 - 2\lambda3 + 1

To factor roots, I set the determinant equal to zero and factored, as follows:
0 = (\lambda3 - 1)2
= (\lambda-1)(\lambda5 + \lambda4 + \lambda3 - \lambda2 - \lambda - 1)
= (\lambda - 1)(\lambda - 1)(\lambda4 + 2\lambda3 + 3\lambda2 + 2\lambda + 1)
= (\lambda - 1)2(\lambda2 + \lambda + 1)2

It has identical real roots at... \lambda<sub>1</sub> = \lambda<sup>2</sup> = 1, and identical complex roots at \lambda<sub>3</sub> = \lambda<sub>4</sub> = 1/2 + \sqrt{3}i/2 and \lambda<sub>5</sub> = \lambda<sub>6</sub> = 1/2 - \sqrt{3}i/2

But the issue is, with a 6 x 6 matrix, which case should I evaluate and how should I go about finding the eigenvectors?

I know complex roots evaluate to the canonical form A\bar{}:
\alpha \beta 0
-\beta \alpha 0
0 0 1

But do I need to evaluate each of the positive and negative complex roots separately, and where do I throw in the \lambda<sub>1</sub> = \lambda<sub>2</sub> canonical form in that big 6 x 6?

Thanks!
 
Hey, buddy are you in sergey nikitin class at ASU?
 
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