What does each matrix do to the unit square?
The columns of a 2×2 matrix are the images of and . Pick a transformation, watch the green image of the unit square, and read off the matrix. These are the OCR-spec transformations.
Reading the OCR spec off the matrix
Rotation anticlockwise: . Reflection in : .
Stretch parallel to an axis: the other axis is invariant and the scale factor sits on the diagonal. Enlargement centre (0,0): , area scale factor . Shear: one axis is fixed; the off-diagonal entry slides points parallel to it.
Rotation by any angle
Drag the angle. The image of A(1,0) lands at (cosθ, sinθ) and the image of B(0,1) lands at (−sinθ, cosθ). Those are the two columns of the rotation matrix.
Doing one transformation then another
If you apply first and then , the single matrix that does both is the product - the first one you apply goes on the right. Order usually matters.
Blue dashed = after . Green = after both. The original unit square is grey.
Invariant points and lines
Invariant means never changing. An invariant point maps to itself. A line of invariant points is a line where every point stays put. An invariant line is a line that maps onto itself (points can slide along it). Type a matrix or load a worked example.
The method
Invariant points: solve , i.e. . If only works it is the single invariant point; if a whole line works it is a line of invariant points.
Invariant lines : the image of must also have gradient . That gives ; each root is the gradient of an invariant line through the origin.