# How can I find all possible Jordan forms?

• I
• laurabon
In summary: 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?
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.

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

Need to use $$(display) or ## (in line) on both ends to bracket Latex expressions here. jedishrfu fresh_42 said: What did you try so far? And what do you know about the characteristic polynomial and its relation to eigenvalues? Thanks for your help . What stops me it's just the beginning. I don't know what the zeroes of T can help me? In general in class I saw how to build jordan form using given matrix. Thanks again mathman said: Need to use$$ (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?

## 1. What is a Jordan form?

A Jordan form is a type of matrix that is used to represent a linear transformation or a matrix. It is named after the mathematician Camille Jordan and is in a special form that makes it easier to analyze and understand.

## 2. How can I determine the Jordan form of a matrix?

To determine the Jordan form of a matrix, you can use a process called diagonalization. This involves finding the eigenvalues and eigenvectors of the matrix and then using them to construct the Jordan form. Alternatively, you can use software or online tools that can automatically calculate the Jordan form for you.

## 3. What are the benefits of using Jordan forms?

Jordan forms have several benefits, including simplifying the analysis of linear transformations and matrices, providing a clearer understanding of the behavior of a system, and allowing for easier computation of powers and inverses of matrices.

## 4. Can all matrices be transformed into Jordan form?

No, not all matrices can be transformed into Jordan form. Only square matrices with complex entries can be transformed into Jordan form. Matrices with real entries may have a different form called real Jordan form.

## 5. Are there any limitations to using Jordan forms?

One limitation of using Jordan forms is that they can only be used for square matrices. Additionally, they may not always be the most efficient or practical way of representing a matrix, and there may be other methods that are better suited for certain applications.

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