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Eigenvectors and Matrices

Date: 09/01/97 at 19:54:34
From: Milenko
Subject: Eigenvectors what is practical use

Dear Dr. Math.

I have been trying for the past two years to understand what is the 
physical presentation of eigenvalues and eigenvectors. 

Can you please give me one good example of a practical use of 
eigenvalues and eigenvector, so I can see what is what and what I can 
do with it?

Thanks in advance.

Date: 09/02/97 at 12:52:08
From: Doctor Ceeks
Subject: Re: Eigenvectors what is practical use


A linear transformation of a vector space is a very common object
with many practical applications (for instance, in random walks,
in three-dimensional computer graphics, and in differential
equation theory).

If you have a linear transformation and want to understand what it 
does, one of the first things you might try to understand is whether
or not there are any vectors that are multiplied by some scalar when 
you apply the linear transformation (i.e. when you apply the linear
transformation, generally, a vector will transform into another vector
of different magnitute AND direction, but there may be some vectors
whose direction is not really affected, though they may be changed to
point the opposite way).

Such vectors are eigenvectors, and the multiplication factor is
the associated eigenvalue.

-Doctor Ceeks,  The Math Forum
 Check out our web site!   

Date: 09/02/97 at 16:24:45
From: Doctor Anthony
Subject: Re: Eigenvectors what is practical use

I will give you a physical description, i.e. using 2 or 3 dimensions 
only, though the ideas can be extended to n dimensions where n is as 
big as you please.

You will be aware that if say a (2x2) matrix M operates on a two-
dimensional column vector v, then that vector is transformed in 
magnitude or direction or both to the vector v'.

    So   M.v = v'

Now for the general (2x2) matrix M there are 2 eigenvalues k1 and k2 
with associated eigenvectors u1 and u2 with the property that:

        M.u1 = k1.u1

        M.u2 = k2.u2

So any point on the vector u1 is transformed to k1.u1 when operated 
upon by M, and similarly any point on u2 will move to k2.u2 after 
transformation by M.  In some problems where M is to transform a 
complicated figure or we wish to describe the transformation clearly, 
it is convenient to use u1 and u2 as the base vectors - i.e. give 
coordinates of all points in terms of u1 and u2 rather than the usual 
(x,y) coordinates, and the transformation matrix then becomes 

        |k1    0|
        |0    k2|

We can find powers of matrices very conveniently using eigenvalues and 
eigenvectors.  It is easy to show that 

  M = (u1 u2)|k1   0|(u1 u2)^(-1)
             |0   k2|

where (u1 u2) is the 2x2 matrix P formed by the columns of u1 and u2.

 Then M^n = P|k1   0|^n P^(-1)
             |0   k2|

      M^n = P|k1^n    0|P^(-1)
             |0    k2^n| 

In probability theory, powers of matrices are frequently required, 
sometimes infinite powers, so some device for handling such a problem 
is clearly very important.

I have touched on a couple of uses of eigenvalues and eigenvectors, 
but they are really part of the basic language of matrix theory, and 
they crop up everywhere matrices are used.  

-Doctor Anthony,  The Math Forum
 Check out our web site!   
Associated Topics:
College Linear Algebra

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