Eigenvectors of recursive set of similarity transformations

Hi!

I need some help with the following:

Consider the sequence

$\displaystyle A_{1} = A \\

A_{2} = P ^{-1}_{1} A_{1} P_{1} \\

A_{3} = P ^{-1}_{2} A_{2} P_{2} \\

. \\

. \\

. ... \\

A_{k+1} = P ^{-1}_{k} A_{k} P_{k} .$

1) Write an expression for the eigenvectors of $\displaystyle A_{k+1}$ using the eigenvectors of $\displaystyle A_{k}$. Then use this to write an expression for the eigenvectors of $\displaystyle A_{k}$ using the matrices $\displaystyle P_{1},..., P_{k-1}$, and the eigenvectors of $\displaystyle A_{1} = A$.

I am completely lost and have no idea where to start. Any pointing me in the right direction would be greatly appreciated!

j

edit: I can't get line breaks to work in the above code...(?)