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

or , as my lecturer likes to write stuff.

The notes then say "It would be easy to compute if were an eigenvector i.e. ". Up to this point, I undersand completely.

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

...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 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.

So basically,where has the came from?Sorry this is a bit long winded by the way, I just thought having the whole question might help.