Suppose we have two sets of orthonormal 16-dimensional vectors , ${x_1,...,x_4,x_5,..,x_8}$ and ${x_1',...,x_4'}$ with angle in-between $x_i$ and $x_i'$ being $\theta$ for 1<=i<=4. Is it possible to find a unitary matrix, $M$, for which $Mx_i = x_i'$ , and also the angle between $Mx_j$ and $x_j$ is $\theta$ for $5<= j<= 8$?
What would be an efficient algorithm to find such an $M$?
Of course, the number 16 and 4 and 8 can be generalized suitably and the answer should not change.