Consider the matrix
$$U(\theta) = M.G(\theta)$$
where
$$G(\theta) = \begin{pmatrix} \cos(2\theta) & -i \sin(2\theta)/\sqrt{2} & -i \sin(2\theta)/\sqrt{2} \\ -i \sin(2\theta)/\sqrt{2} & \cos^2(\theta) & -\sin^2(\theta) \\ -i \sin(2\theta)/\sqrt{2} & -\sin^2(\theta) & \cos^2(\theta) \\ \end{pmatrix}$$
and
$$M = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$
I define the repeated matrix product $P(n,\theta) = U^n$ (i.e. $P(3,\theta)=UUU)$.
I need to calculate the result of an infinite amount of these products, as the angle $\theta$ also goes to zero, but whilst the product $n\theta=\phi$ is a constant. So I need to find
$$R = \lim_{n \rightarrow \infty} P(n,\phi/n)$$
I'm really hoping I can express the matrix elements as some kind of known function(s). Please could you give me some advice on how to proceed.