Eigenvectors

Date: 05/13/97 at 22:50:57
From: Nader Iskander
Subject: Eigenvectors

I have a question on determining any kind of pattern related to 
eigenvectors. First of all, I do not fully understand what 
eigenvectors are. All I know is that they can be put in matrix form, 
and when mutiplied by a point, give a scalar multiple of the original 
point. If there is more to them, can you please tell me? 

Thank you very much.

Date: 05/14/97 at 16:44:53
From: Doctor Anthony
Subject: Re: Eigenvectors

The eigenvectors of a matrix are vectors that do not vary in direction 
when transformed by the matrix.  A point on an eigenvector can move up 
or down the vector when transformed by the matrix, but it will remain 
on that vector. The scale factor by which such a point moves is given 
by the associated eigenvalue.

If v is an eigenvector and k the associated eigenvalue, then for 
matrix M, we have: 

                Mv = kv

A 3 x 3 matrix will, in general, have 3 linearly independent 
eigenvectors, each with its associated eigenvalue. If we use these 
three vectors as the base vectors of the 3D space (that is the 
position of any point is given in terms of u, v, w the eigenvectors), 
then on transforming by matrix M, the new position is simply k1*u, 
k2*v, k3*w  where k1, k2, k3 are the 3 eigenvalues, associated 
respectively with u, v, and w.

The eigenvalues and then the eigenvectors are found by solving:

           |M-k*I| = 0      

For a 3 x 3 matrix there will be three roots k1, k2, k3. Using these 
values in turn in the equation Mv1 = k1*v1, we obtain the three 
eigenvectors.

There are many uses of eigenvectors, a particularly important one is 
in finding powers of matrices. For example, M^200 is not to be 
lightly undertaken, but by using the diagonal form of the matrix (in 
which eigenvalues and eigenvectors play a key role) we can write down 
the result in a few minutes.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/