I Correlation Matrix of Quadratic Hamiltonian

[thatboi]

I am struggling to rederive equations (61) and (62) from the following paper, namely I just want to understand how they evaluated terms like $\alpha\epsilon\alpha^{T}$ using (58). It seems like they don't explicitly solve for $\alpha$ right?

 

[DAntanov]

First off, very neat paper. As for (58), if they are reabsorbing the matrix of phases (U) into the definition of α, then anything having to do with α is dependent on what U is set to (in this case = 1). I think this is informed by (51) and the instructions for (60). You are correct in saying they dont explicitly solve for α, but they could have shown their work a bit more before (60), those instructions are unnecessarily packed.