Uniqueness of the modal matrix

Question

Suppose that matrix $A$ is diagonalizable, i.e. there exists an invertible matrix $P$ and a diagonal matrix $D$ so that $P^{-1}AP=D$. We know that (i) we can form $P$ from eigenvectors of $A$ and (ii) $P$ and $D$ are not unique as we can change their column.

My questions are that:

1. Is it correct that the columns of $P$ must be eigenvectors of $A$?
2. If I set the constraint that all diagonal elements of $P$ are $1$, then is $P$ unique?

Answer 1

1. Yes, that is correct.
2. No. Suppose that $A=D=\operatorname{Id}$. Then if $P$ is any invertible matrix, you will have $P^{-1}AP=D$.