Question

I need to show an example (2x2) , that similar matrices can have different eigenvectors

Answer 1

What do you mean by similar matrices?

Answer 2

Matrix A similar to matrix B
P-1AP = p-1BP

Answer 3

Simply try two matrices A and P. I tried\[A=\left[\begin{matrix} 1&2\\3&4\end{matrix}\right]\]\[P=\left[\begin{matrix} 4&3\\2&1\end{matrix}\right]\]And it seems to spit out different eigenvectors for \(A\) and \(B=P^{-1} A P\) I'd suggest you check this yourself though.

I didn't get very pretty eigenvectors either, so you might want to try a simpler matrix.