How can I find all possible Jordan forms?

[laurabon]

TL;DR Summary

find all possible Jordan forms

Hi this is my first message in this forum , I have this problem in my linear algebra course and I have never seen this type. Let $T : \mathbb{Q}^3 → \mathbb{Q}^3 $ a linear application s.t $(T^7 + 2I)(T^2 + 3T + 2I)^2 = 0$ Find all possible Jordan forms and the relative characteristic polinomial . Thanks to anyone for the help.

 

[fresh_42]

What did you try so far? And what do you know about the characteristic polynomial and its relation to eigenvalues?

 

[mathman]

Need to use $$ (display) or ## (in line) on both ends to bracket Latex expressions here.

 

[laurabon]

What did you try so far? And what do you know about the characteristic polynomial and its relation to eigenvalues?

Thanks for your help . What stops me it's just the beginning. I don't know what the zeroes of T can help me? In general in class I saw how to build jordan form using given matrix. Thanks again

 

[laurabon]

Need to use $$ (display) or ## (in line) on both ends to bracket Latex expressions here.

Thanks , next time i'll use them

 

[DaveE]

Like this:
Let $$T : \mathbb{Q}^3 → \mathbb{Q}^3 $$ a linear application s.t $$(T^7 + 2I)(T^2 + 3T + 2I)^2 = 0$$

Easy to read questions get more traction. Seriously, it wasn't THAT hard to fix.

 

[fresh_42]

Thanks for your help . What stops me it's just the beginning. I don't know what the zeroes of T can help me? In general in class I saw how to build jordan form using given matrix. Thanks again

You should start to factorize your equation. Does $T^7-2$ have rational roots? What are the roots of the other factor? This should tell you something about the eigenvalues.

 

[laurabon]

You should start to factorize your equation. Does $T^7-2$ have rational roots? What are the roots of the other factor? This should tell you something about the eigenvalues.

I think that I found the correct solution to this question . The possible minimal polynomials are $$(x+1)^a(x+2)^b$$ with any $$0≤a≤2, 0≤b≤2, 1≤a+b≤3$$​

now what about characteristic polynomial ?I only need this to find the jordan form am I right?​

 

[fresh_42]

I think that I found the correct solution to this question . The possible minimal polynomials are $$(x+1)^a(x+2)^b$$ with any $$0≤a≤2, 0≤b≤2, 1≤a+b≤3$$​

now what about characteristic polynomial ?I only need this to find the jordan form am I right?​

Yes.

What you have is almost the minimal polynomial, and the minimal polynomial divides the characteristic polynomial. Now, what about the factor $T^7+2I$? What do we know about it?