A matrix is diagonalizable, so what? I mean, you can say it's similar to a diagonal matrix, it has $n$ independent eigenvectors, etc., but what's the big deal of having diagonalizability? Can I solidly perceive the differences between two linear transformation one of which is diagonalizable and the other is not, either by visualization or figurative description?
For example, invertibility can be perceived. Because non-invertible transformation must compress the space in one or more certain direction to $0$. Like crashing a space flat.

Diagonalization is shearing any source of origin  the x directional and  y directional matric elements of rows and  column from a source of for example radiation of light rays into squeezing along hypotenuse out of a right angled  triangle sides as such resultant a+ib module of shearing.This happens  as a result of stretching.It really bends a light ray by its refractive index along input and output planes.On the line of thought this may play the role of light ray invisibility by the Einstein gravity of bending typically applicable in Quantum mechanics.A sort of squeezing or shearing along hypotenuse side.When all the elements  are diagonally shifted the potential becomes  zero and not so when more than zero above the diagonalization of directional matrics elements.This may also be called a twisting along axial planes as a function of twisting angle of ray transfer matrics  pave the way for  magnification as divergence ,convergence as well as for invisible clocking dynamics using laser beams.
