Initial conditions on $\mbox{$\mathbf{P}$}$ and $\mbox{$\mathbf{Q}$}$

Generally speaking, the only way to ensure that the matrices $\mbox{$\mathbf{P}$}(s)$ and $\mbox{$\mathbf{Q}$}(s)$ are complex, being $\mbox{$\mathbf{C}$}(s)$ a real matrix, is by selecting a proper choice of complex initial conditions for both matrices. As shown by the literature such proper choice is a matter of intense debate, seemingly not yet solved. There are however particular choices of real $\mbox{$\mathbf{P}$}(0)$ and $\mbox{$\mathbf{Q}$}(0)$, which are relevant for the discussion of Gaussian beams. Due to such relevance those choices are presented in this section for the case of a point source and for the case of a linear source.