Are only normal matrices diagonalizable?

Question

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

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

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

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Does this mean that only normal matrices are diagonalizable?

Answer 1

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).