Eigenvalues and Diagonalization

Question

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.

Answer 1

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.