Jordan Basis for Matrix A: Missing Something Trivial?

[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?

 

[micromass]

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?

 

[estro]

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?

 

[micromass]

I think it's good to start with wiki: http://en.wikipedia.org/wiki/Jordan_normal_form

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.

 

[estro]

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.

 

[micromass]

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!

 

[estro]

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]

 

[micromass]

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]

Note that your matrix already is in Jordan normal form...
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.

 

[estro]

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?

 

[micromass]

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