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Complexity

Member

From: Denmark

Registered: 2013-12-27

Posts: 16

Matrices that are diagonalizable

Hello honored all-knowing math-masters!

I am farely new to this level of math so I hope that you guys could come up with a fast simple answer to this question:
Does a special type of matrix, that is always diagonalizable, exist? By type I'm referering to stochastic, irreducible ect.

Reason:
I am currently working with Google PageRank in which the so called PowerMethod is used. According to

http://www.cs.cornell.edu/~bindel/class … /lec26.pdf

the PowerMethod will make a vector converge. In the Google case I've been using stochastic matrices and for these I've proven 1 to always be the largest (absolute) eigenvalue, hence the convergence is assured according to the link, but only if the stochastic matrix is diagonalizable.
The matrix that's being dealt with is row-stochastic & irreducible/primtive. Is any of these types making the matrix diagonalizable?

I've already read something about hermitan and unitary matrices should be diagonalizable, but they don't seem to be completely the same 'kind' as the matrices I'm working with.

Thank you from Denmark (Hope my english is somewhat understandable).

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices that are diagonalizable

Hi;

Please give a better link to that pdf.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Complexity

Member

From: Denmark

Registered: 2013-12-27

Posts: 16

Re: Matrices that are diagonalizable

Sorry bobbym but I get this message everytime i try to write the full path: "Sorry. In an effort to stop automated spam only established members can post links. Please describe where instead." (3 w's should do the trick to get on though).

Last edited by Complexity (2013-12-27 10:31:00)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices that are diagonalizable

I have fixed the link for you and have the pdf.

Some rules are if a matrix has distinct eigenvalues it is diagonalizable.

Have you been here:

http://en.wikipedia.org/wiki/Diagonalizable_matrix

The power method like all iterative methods is tricky, and sometimes will converge even if some of the conditions are not met.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Complexity

Member

From: Denmark

Registered: 2013-12-27

Posts: 16

Re: Matrices that are diagonalizable

Yes the triple-w should be enough.

Hmmm it's not possible to say that all the eigenvalues are distinct (I would say). Oh I forgot a very important thing: I'm looking at left eigenvectors. So the eigenvalue problem looks as the last part of this:
[math]Ax=\lambda x \Leftrightarrow x^T(A-I)^T = x^T(A^T-I^T) = 0[\math]
It doesn't matter thinking of eigenvalues, because I've proven these are the same for the left and right eigenvectors. I think I will look at what it takes to only have distinct eigenvalues (I don't have a proof that shows that all eigenvalues are distinct).
Thanks for a response

Edit: Oh didn't see your post there! Yes I have been to that page of wikipedia, since I needed this to understand the proof. I though only have looked for some understandings of the proof! It's getting late here so I will stop the search for tonight, but thanks for the fast response, didn't expect that

Last edited by Complexity (2013-12-27 10:56:09)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices that are diagonalizable

Apparently most matrices are diagonalizable but it is difficult to find the conditions that allow it.

All I know about is:

1) Not all stochastic matrices are diagonalizable.

2) Hermitian matrices are diagonalizable.

3) An nxn matrix A is diagonalizable over the field F if it has n distinct eigenvalues in F. But even if it does not have distinct eigenvalues it may be diagonalizable.

4)As a rule of thumb, over C almost every matrix is diagonalizable.

5) Symmeteric matrices are diagonalizable.

You can get a little more from here too:

http://www.sosmath.com/matrix/diagonal/diagonal.html

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.