If two matrices are similar, can they be diagonalized? In other words, can diagonalize the matrix and its similar matrix? Best examples

If two matrices are similar, can they be diagonalized? In other words, can diagonalize the matrix and its similar matrix? Best examples

Of course not
Example: a=
1 1
0 1
For any invertible matrix P, P ^ - 1AP is similar to a, but they cannot be diagonalized