diagonalisability of matrix few properties

Solution 1:

Let $A\in \mathcal M_{n\times n}(\Bbb C)$.

Some common criteria are:

1. If $A$ has $n$ distinct eigenvalues, then $A$ is diagonalizable.
2. $A$ is diagonalizable if, and only if, the sum of the geometric multiplicties of all the eigenvalues equals $n$. (Note that 1. is a particular case of this).
3. $A$ is normal if, and only if, $A$ is unitarily diagonalizable.
4. $A$ is hermitic if, and only if, $A$ unitarily similar to a diagonal matrix with only real entries.
5. $A$ is unitary if, and only if, $A$ is unitarily similar to a diagonal matrix which entries on the main diagonal have absolute value equal to $1$.

Solution 2:

A matrix is diagonalizable iff its minimal polynomial has distinct roots and factors into linear factor.

Solution 3:

You can see diagonalisability of matrix few properties in Wikipedia, by coincidence this article has the same title as your question.