Given vectors $\boldsymbol{a}$ and $\boldsymbol{b} \ne \boldsymbol{0}$ in the same Hilbert space, does
$$\boldsymbol{\Theta a} = \lambda \boldsymbol{b}$$
always hold for some unitary $\boldsymbol{\Theta}$ and real scalar $\lambda$?
If the answer is yes and there are multiple solution pairs, how to find $\boldsymbol{\Theta}_\max$ that corresponds to $\lambda_\max$?
The problem looks simple but the closest I have in mind is the generalized eigenvalue problem. Thank you for any hint!
Edit: Sorry my description was misleading. I made some very naive attempts (1) using pseudo-inverse and (2) on the 2x2 case where the unitary matrix has a closed-form expression.