We know that normal matrices are diagonalizable, but the converse is not true. For example, see here. Since a diagonalizable matrix represents a scaling operation under certain basis, so I wonder what additional geometrical meanings a normal matrix processes to be distinguished from other diagonalizable matrices. In other words, how to geometrically interpret matrices that can be diagonalized by unitary and non-unitary matrices? Thanks!
So, eigenvectors are the axes where there is a magnification/contraction.
For normal matrices, these axes are orthogonal to each other, thus, a matrix is normal if and only if it represents a linear transform that scales the coordinate axes, (in a suitable chosen orthogonal basis).
A diagonalizable matrix in general is a similar linear transform, (scaling of the coordinate axes), but the basis need not to be orthogonal.