I am asked to find the eigenvalues and eigenvectors of the matrix:

I find them as , and ,

The equations I need to solve are: and . These equations are coupled (whatever that means...)

I turn these into a pair of simultaneous equations sort of thing to form the matrix equation , where X' is and

As far as I know, I have diagonalise A which will give me . I know and using the eigenvalues.

So I go along...

Subbing matrices in...

Can someone check if what I've done is correct so far? And if so, am I allowed to expend all the differentials and all the matrices then treat them as simultaneous equations. Also, I'm not too sure how to get rid of the when there is no variable in the equations...