🚀 go-pugleaf

Alan,

Many many thanks for this.

I got to the point where I worked out that I could calculate the
polynomial, and can calculate the roots for this.  I just didn't know
how to get the eigenvectors from this.

e.g. [ 2 1 ]
     [ 3 4 ]

gives the polynomial x^2 - 6x + 5 = 0
giving roots (x - 1) (x - 5)

As you state,

> The eigenvectors can then be written as (b, x-a) or (x-d, c).

Your solution provides me with 4 potential eigenvectors.

(1, 1-2) = (1, -1) IS an eigenvector for the eigenvalue 1.
(1, 5-2) = (1, -3) IS NOT an eigenvector for the eigenvalue 5.
(1-4, 3) = (-3, 3) IS NOT an eigenvector for the eigenvalue 1.
(5-4, 3) = (1, 3) IS an eigenvector for the eigenvalue 5.

However, it is a simple matter to work out which 2 work, and discard
the others.

Problem solved.  Thanks again

Jon








"Alan Miller" <amiller@bigpond.net.au> wrote in message news:<iB8G7.310759$bY5.1235789@news-server.bigpond.net.au>...
> jonmach wrote in message ...
> >Hi All,
> >
> >I've trawled the net and have found some very comprehensive libraries
> >and source for calculating the eigenvalues/eigenvectors of an nxn
> >matrix.  However, this is very much overkill for what I need.
> >
> >I have a single 2X2 matrix holding real values and simply need to
> >calculate the 2 eigenvalues and corresponding eigenvectors.  Is there
> >a very simple algorithm for this that already exists in
> >C/Fortran/Java/C++?
> >
> >Many many thanks in advance for any assistance on this.
> >
> >Jon
> 
> The matrix:-
> ( a  b )
> ( c  d )
> has eigenvalues obtained by solving (a-x)(d-x) - bc = 0 for x.
> This is simply a quadratic equation, and the solutions are:
> 2x = -(a+d) +/- sqrt[ (a-d)^2 + 4bc ]
> The roots can of course be complex, if b & c have opposite signs
> and are moderately large compared with the diagonal elements a & d.
> The eigenvectors can then be written as (b, x-a) or (x-d, c).
> 
> Complex operations and variables are intrinsic in Fortran, so that
> may be the easiest language to use.
> 
> Now I have done your homework for you, you may like to look up
> the numerical solution of quadratic equations to minimize cancellation
> errors.   If (a-d)^2 is approximately equal to -4bc, the cancellation errors
> could be very large, if you use the simple formula above.