Are only normal matrices diagonalizable? 
*

*For a square $n\times n$ matrix to be diagonalizable, it needs to have $n$ linearly independent eigenvectors.

*If and only if a matrix is normal can the $n$ eigen-vectors be made to form an orthonormal basis.

*$n$ linearly independent vectors can always be made to form a orthonormal basis

Does this mean that only normal matrices are diagonalizable? 

All normal matrices are diagonalizable. Not all diagonalizable matrices are normal.
Try to find an example of a diagonalizable but not normal matrix on your own (say, in 3 x 3 matrices). 
