# Jordan Basis for Matrix A: Missing Something Trivial?

• estro
In summary: Yes, it can be time-consuming, but it's not impossible. You could try using a computer program to find the basisses, or you could try a more brute force approach and try to find all the eigenvalues and eigenvectors yourself.Thanks!
estro
In my book I see that the author finds the jordan basis for matrix A={(-3,9),(1,3)} by finding nullspace for (A-3I) and (A-3I)^2 without any justification, do I miss something trivial here?

estro said:
In my book I see that the author finds the jordan basis for matrix A={(-3,9),(1,3)} by finding nullspace for (A-3I) and (A-3I)^2 without any justification, do I miss something trivial here?

Hi estro!

Your book just tries to find the generalized eigenvectors of the eigenvalue 3. Your book should mention what generalized eigenvectors are and why they are important though...
What book are you using?

Thank you for the quick reply!

I'm an Israeli student so I'm pretty sure you won't be familiar with the book. [It's called Linear Algebra 2 from the open university]
Can you refer me to a source where I can read and clear my concerns regarding this idea?

Check the section on "Complex matrices" which explains what generalized eigenvectors are and why they are important.
Also, check out the "Example" section which gives an example on how to calculate the Jordan normal form.

Thank you! I should have been looking there in the first place=...)
If you don't mind I'll ask here again if wikipedia won't be enough for me.

estro said:
Thank you! I should have been looking there in the first place=...)
If you don't mind I'll ask here again if wikipedia won't be enough for me.

Feel free to ask anything you don't understand!

After reading the wiki and the proof for the existence of the jordan form I think that I'm getting into the idea, however I was able to think about the following example:

Lets choose matrix A={(2,0,0),(0,0,1),(0,0,0)} so the characteristic polynomial is also the minimal: p(t)=(t-2)t^2.

Now this is how I find the jordan basis:
1. NullSpace(A-2I)=Sp{(1,0,0)}
2. NullSpace(A-0I)=SP{(0,1,0)}
3. NullSpace(A-0I)^2={(0,0,1),(0,1,0)}

So the jordan basis is {(1,0,0),(0,1,0),(0,0,1)}
But how can I in what order to write these column vectors in my matrix P? [to satisfy P^{-1}AP]

estro said:
After reading the wiki and the proof for the existence of the jordan form I think that I'm getting into the idea, however I was able to think about the following example:

Lets choose matrix A={(2,0,0),(0,0,1),(0,0,0)} so the characteristic polynomial is also the minimal: p(t)=(t-2)t^2.

Now this is how I find the jordan basis:
1. NullSpace(A-2I)=Sp{(1,0,0)}
2. NullSpace(A-0I)=SP{(0,1,0)}
3. NullSpace(A-0I)^2={(0,0,1),(0,1,0)}

So the jordan basis is {(1,0,0),(0,1,0),(0,0,1)}
But how can I in what order to write these column vectors in my matrix P? [to satisfy P^{-1}AP]

The order in which to write the column vectors doesn't matter, you just need to group the vectors from the same eigenspace together. So you could write

$$P=[(1,0,0),(0,1,0),(0,0,1)]$$

or

$$P=[(0,0,1),(0,1,0),(1,0,0)]$$

the order of the vectors will only induce a permutation of the Jordan blocks, and that won't matter.

Thanks!
But did I understand the concept of finding jordan basis? []
Will I be able to find the jordan basis for every possible matrix with this technique?

estro said:
Thanks!
But did I understand the concept of finding jordan basis? []
Will I be able to find the jordan basis for every possible matrix with this technique?

Yes, you understood the technique. Take a look at http://www.google.be/url?sa=t&sourc...g=AFQjCNEs8yAwLNlJ4PCC-tbXajtjSyupdw&cad=rja" for more examples of the Jordan normal form. If you can do all those examples, then you understand the concept well!

But finding Jordan basisses is a very time-consuming thing for large matrices

Last edited by a moderator:

## 1. What is the Jordan Basis for Matrix A?

The Jordan Basis for Matrix A is a set of vectors that spans the entire space of a given matrix A. These vectors are the generalized eigenvectors of A and can be used to diagonalize A.

## 2. How is the Jordan Basis for Matrix A calculated?

The Jordan Basis for Matrix A is calculated by finding the eigenvalues and eigenvectors of A and then finding the generalized eigenvectors by solving a system of equations.

## 3. Why is the Jordan Basis for Matrix A important?

The Jordan Basis for Matrix A is important because it allows us to easily find the Jordan form of A, which is useful for solving systems of linear equations and understanding the behavior of A.

## 4. Can the Jordan Basis for Matrix A be used for any matrix?

Yes, the Jordan Basis for Matrix A can be used for any square matrix. However, it is most useful for matrices with repeated eigenvalues.

## 5. What happens if the Jordan Basis for Matrix A is missing a trivial vector?

If the Jordan Basis for Matrix A is missing a trivial vector, it means that A is not diagonalizable. This could happen if A has a repeated eigenvalue but not enough generalized eigenvectors to span the space. In this case, the Jordan form of A will have blocks of size larger than 1 along the diagonal.

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