I How can I find all possible Jordan forms?

laurabon
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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.
 
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What did you try so far? And what do you know about the characteristic polynomial and its relation to eigenvalues?
 
Need to use $$ (display) or ## (in line) on both ends to bracket Latex expressions here.
 
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fresh_42 said:
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
 
mathman said:
Need to use $$ (display) or ## (in line) on both ends to bracket Latex expressions here.
Thanks , next time i'll use them
 
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.
 
laurabon said:
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.
 
fresh_42 said:
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?​
 
laurabon said:
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?
 
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