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: $$\begin{bmatrix} 0.9998 & -0.0175 \\ 0.0175 & 0.9998 \end{bmatrix}$$
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.