Originally Posted by
CaptainBlack Let $\displaystyle \bold{A}$ be a matrix with an eigen value of absolute value greater than that of its other eigen values, then the ittereation:
$\displaystyle \bold{b}_{k+1}=\frac{\bold{Ab}_{k}}{|\bold{Ab}_k|} \ \ \ \ \ ...(1)$
converges to the eigen vector corresponding to the eigen vector of largest absolute value.
If $\displaystyle \bold{b}$ is this limit the largest eigen value can be found from any component of the equation:
$\displaystyle \bold{Ab}=\lambda \bold{b} \ \ \ \ \ \ \ ...(2)$
So you start with a random vector $\displaystyle \bold{b}_0$ and apply the itteration in $\displaystyle (1)$ to it untill a satisfactory lefel of convergence is observed, then use any non-zero component of $\displaystyle \bold{b}$ and equation $\displaystyle (2)$ to find the eigen value.
Now this requires a lot of computation and I expect you are supposed to use some sort of tool to perform the itteration.
RonL
This can be mechanised as in the following:
Code:
>A=[4,1,-1;2,3,-1;-2,1,5]
4 1 -1
2 3 -1
-2 1 5
>
>function itterate(A)
$ bm=random(3,1);
$ repeat
$
$ bp=A.bm;
$ bp=bp/sqrt(sum((bp^2)'));
$ l=A.bp/bp;
$ lambda=l(1);
$ if totalmax(abs(bm-bp))<1e-6
$ break
$ endif
$ bm=bp;
$ end
$ return {lambda,bp}
$endfunction
>
>{lambda,b}=itterate(A);lambda," ",b
6
0.577351
0.577351
-0.577349
>
RonL