The typical definition given for a diagonalizable matrix is:
Given $A\in M^F_{n\times n}$
$A$ is diagonalizable $\iff$ A has $n$ linearly independent eigenvectors.
Is it also true that
$A$ is diagonalizable $\iff$ A has $n$ unique eigenvalues.
The typical definition given for a diagonalizable matrix is:
Given $A\in M^F_{n\times n}$
$A$ is diagonalizable $\iff$ A has $n$ linearly independent eigenvectors.
Is it also true that
$A$ is diagonalizable $\iff$ A has $n$ unique eigenvalues.
No. It is not true that $A$ is diagonalizable $\iff$ A has $n$ unique eigenvalues.
For instance, the identity matrix is diagonalizable but all eigenvalues are the same, namely $1$.
However one way in your claim is true i.e.
A has $n$ unique eigenvalues $\implies$ $A$ is diagonalizable.