Animating a smooth linear transformation After watching this video on matrices, I tried to make a simple animation of 2D linear transformations. Starting with a set of 2D points in a grid, I applied a 2x2 matrix to each point and obtained a set of transformed points.

I'm not sure how to animate the transition between the initial and final states. At first, I made each point move along the straight path between its initial and final position. This caused problems for rotation matrices. For example, a 180$^{\circ}$ rotation should show the points rotate about the origin in circular arcs, but my method made it look like the whole grid flipped around, without rotating. 

A better rotation animation can be made by applying a very small rotation, say 1$^{\circ}$, multiple times. The matrix for a 1$^{\circ}$ rotation is close to the identity matrix:
$$\begin{bmatrix} 0.9998 &amp; -0.0175 \\ 0.0175  &amp;  0.9998 \end{bmatrix}$$

Is there some general formula that can generate a differential transformation matrix, given the full transformation matrix? I would like it to work for any 2x2 matrix, without needing to specify the type of transformation.
As a general rule, you cannot interpolate transformation matrices. In stead, you decompose them into their individual values, then interpolate those and recompose. 

The Möbius transformation as suggested in the comments sounds interesting, but tradionally I'd just extract scale and rotation and interpolate those.

Assuming a transformation matrix

    | a  b |
T = | c  d |



the scale is [sqrt(a² + c²), sqrt(b² + d²)]
The rotation angle is atan(c/a)


You'd now interpolate this scale and rotation angle, recomposing the matrix at each frame.

[Edit] Corrected the angle (as spotted by Nathan Reed) [/Edit]