Theorem 1: An nxn matrix A is diagonalizable iff it has n linearly independent eigenvectors, and these form P.

Theorem 2: If A has n distinct eigenvalues then the corresponding eigenvectors are linearly independent and A is diagonalizable.

Careful with Theorem 2 - not iff! A can still have less than n distinct eigenvalues and still have n linearly independent eigenvectors to be diagonalizable.