# Matrix generator / representation matrix for equal spacing along a chirp function?

This question is similar to several sets of previous questions dating back to when I joined the site 1, 2, 3. This time it regards a sine function of quadratic frequency. a.k.a. a "chirp signal". I know for the simpler case of constant frequency, we can use a generator which is infinitesimal rotation matrix:

$$R = \begin{bmatrix} \cos(\lambda\epsilon) & \sin(\lambda\epsilon)\\-\sin(\lambda\epsilon)&\cos(\lambda\epsilon) \end{bmatrix}$$

we can plot $$\lambda \epsilon$$ vs $$\sin(\lambda \epsilon)$$ and get any constant $$\lambda$$ frequency sine. $$({R^k})_{1,2} = \sin(\lambda\epsilon k)$$

But what about quadratic frequency? Like plotting $$x$$ vs $$\sin(\lambda x^2)$$. How can we create a generator matrix in the same sense for this situation?