Geometrically, what is the difference between a diagonizable matrix and an orthogonally diagonizable matrix?
I understand the difference algebraically, as explained here and many other places. But I'm struggling to see the difference geometrically: both will not rotate (assuming the field is the reals), but will stretch by different amounts along different axes (Wikipedia calls this a inhomogenous dilation). Graphically, I can't see the difference between, say, a circle or ellipse dilated inhomogenously along orthogonal axes versus non-orthogonal axes: in either case, it will remain an ellipse.
In fact: There is no geometric difference! Every plane transform of a diagonizable matrix is geometrically congurent to an orthogonally diagonizable matrix, via the polar decomposition. See Conjecture: Every shear transformation of the plane is congruent to a dilation .
(I'm using congruence in its geometric meaning: $F$ is congruent to $F'$ if there is an isometry sending $F \mapsto F'$.)