# Geometrical meanings of diagonalizable and normal matrices

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!

• Diagonalizing Matrix for Normal matrices is a orthogonal matrix. Orthogonal matrices represent rotation. Thus Normal matrices first rotates your vector to their co-ordinate system (given by the columns), then scales the components of rotated vector, and then rotates it back. Note that this co-ordinate system is also orthogonal. But for other matrices, the co-ordinate system in which scaling happens is not a orthogonal co-ordinate system. – dineshdileep Jan 10 '13 at 7:16
• @dineshdileep, I wonder if "orthogonal matrix/matrices" in your description should be relaxed to "unitary matrix/matrices". – Computist Jan 15 '15 at 1:36