**2.3. Solving the system of statistical equilibrium equations**

Let us now proceed to set the system of equations to calculate the relative population ratios.

For notation purposes, let us define:

so that eq. (2) reads (taking into account the possible effect of fluorescence):

or equivalently in matrix notation:

After rewriting the system to have the solution expressed in terms of
the populations
relative to the ground level *n*_{1} we are left with:

Eq. (18) is the linear system of equations that is solved by PopRatio.