Matrix iterations

Urm... I'm not so sure. For example if you repeatedly apply the matrix $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ to the vector $(1,0)$ then you end up in an endless loop of (1,0) -> (0,1) -> (-1,0) -> (0,-1) -> (1,0) and so on.

In general, the matrix will have a basis of eigenvectors, and one will have an eigenvalue with modulus strictly bigger than the others (this occurs with probability 1) (*). Call this eigenvector v.
If you then choose a random initial vector and represent it using the eigenvector basis, then with probability 1, the coefficient of v will be non-zero (**).
Then repeatedly multiplying by the matrix causes that coefficient to grow to dominate all others, and so the vector tends towards the eigenline corresponding to the biggest eigenvector.

If (*) or (**) turn out not to be true (because you are unlucky, or because you have an especially fiendish examiner (in which case I guess you're still unlucky)), then all bets are off. (For any specific case, it's unlikely to be that hard to work out what happens, but a general solution/description is going to be tricky).

I

But that matrix has no real eigenvalues. I am talking about a matrix with two.




Right, I see. Thanks. So it basically will tend towards the eigenline corresponding to the eigenvalue with the greatest modulus.
(tried to rep you but it won't let me)

There are also similar iterative methods for finding the eignevalue of smallest magnitude or the eigenvalue nearest to a particular value.

hmm...hadn't really thought of using iteration to actually find the eigenvalue... I usually just work them out.

But that matrix has no real eigenvalues. I am talking about a matrix with two.

What about the eigenvalues of a 20x20 matrix?

Iterative processes definately have their place although we have computers these days which takes a lot of the pain out of it

Ah, sorry, I didn't know you were working over $\mathbb{R}$. Anyway it seems DFranklin has answered your question

yes, I should have been more precise. As I am still a child (mathematically) I only tend to paddle in the safe, shallow waters of $\mathbb{R}$

yes, I should have been more precise. As I am still a child (mathematically) I only tend to paddle in the safe, shallow waters of $\mathbb{R}$

Funny you should say that -- many things involving eigen{anything} and matrices are made a lot more simple if you work over $\mathbb{C}$!

I know I perhaps won't understand, but care to explain for someone curious?

Funny you should say that -- many things involving eigen{anything} and matrices are made a lot more simple if you work over $\mathbb{C}$!

Well for example, there are lots of matrices that you can't diagonalise over $\mathbb{R}$ (e.g. almost all rotation matrices) which you can diagonalise over $\mathbb{C}$, and working over $\mathbb{C}$ we get nice things like Jordan normal form. Basically, because much of the theory of matrices relates to the theory of polynomials, things become easier to do if you work in an algebraically closed field (which $\mathbb{R}$ is not).

interesting because in my FP3 module we have to diagonalise symmetric matrices and I always wondered if you could diagonalise other matrices and what the use of that was.