diagonalized

Expressing the eigenvectors of the diagonalized matrix in the basis :

These eigenvalues are the values that will appear in the diagonalized form of matrix A, so by finding the eigenvalues of A we have diagonalized it.

Note there is no preferred order of the eigenvectors in P; changing the order of the eigenvectors in P just changes the order of the eigenvalues in the diagonalized form of A.