Find some family of transformation with the given constraints.

Question

I need to find some family of transformations that satisfies the following constraints:

1. Every transformation must be linear
2. All nonzero vectors transformed by all transformation in this family are eigenvectors
3. Given some transformation in the family satisfying a -> a' and d -> d', a' and b' must be related by some function f and d' and c' must be related by the same function f. This function can not change in the family.

Picture of desired transformation

Is there any family of transformations which satisfies all of these conditions? If so what is it? If not why not? I can find a solution that satisfies any two conditions, but I can't seem to find one that satisfies all three.