Okay, I'll write out the question in full, but my problem isveryspecific to something that crops up later on in the solution I have here. Sorry if this is in the wrong subforum, but the module's called "Linear Methods", so I thought this seemed appropriate.

Basically it's a method to compute large power of matrices, and the example gives

$\displaystyle \left(\begin{array}{cc}0.9 & 0.2 \\ 0.1 & 0.8\end{array}\right)^n\left(\begin{array}{c}x_1 \\ x_2\end{array}\right)$ or $\displaystyle \underline{\underline{A}}^n\underline{v}$, as my lecturer likes to write stuff.

The notes then say "It would be easy to compute $\displaystyle \underline{\underline{A}}^n\underline{v}$ if $\displaystyle \underline{v}$ were an eigenvector i.e. $\displaystyle \underline{\underline{A}}^n\underline{v}=\lambda^n \underline{v}$". Up to this point, I undersand completely.

Then we start finding the eigenvectors and the calculation goes as such

$\displaystyle \left(\begin{array}{cc}0.9 & 0.2 \\ 0.1 & 0.8\end{array}\right)\left(\begin{array}{c}1 \\ -1\end{array}\right)=\left(\begin{array}{c}0.9\time s1-0.2\times1 \\ 0.1\times1-0.8\times1\end{array}\right)$

$\displaystyle =\left(\begin{array}{c}0.7 \\ -0.7\end{array}\right)=0.7\color{green}\boxed{\left (\begin{array}{c}1 \\ -1\end{array}\right)}$

...and he notes that the green box is always an eigenvector. So far, I still understand. It's this next bit that's confused me. He notes that $\displaystyle 1$ is always an eigenvalue, but he just seems to have plucked this vector out of the air (the one in the red box) and I really can't see where it's came from. It's probably very simple butthisis my problem with the example/solution.

$\displaystyle \left(\begin{array}{cc}0.9 & 0.2 \\ 0.1 & 0.8\end{array}\right)\color{red}\boxed{\left(\begi n{array}{c}2/3 \\ 1/3\end{array}\right)}\color{black}=1\left(\begin{ar ray}{c}0.9(2/3)+0.2(1/3) \\ 0.1(2/3)+0.8(1/3)\end{array}\right)$

$\displaystyle =1\left(\begin{array}{c}2/3 \\ 1/3\end{array}\right)$

So basically,where has the $\displaystyle \mathbf{\left(\begin{array}{c}2/3 \\ 1/3\end{array}\right)}$ came from?Sorry this is a bit long winded by the way, I just thought having the whole question might help.