Thank you for your time.
I played some more with my function and came to the following (nothing fancy, just using property of trace):
$$tr[(G^TG)^{-1}] = tr[G^{-1}G^{-T}] = tr[AA^T] = \sum_{i,j}A^2_{ij} = \sum_{i,j}G^{-2}_{ij}$$
Once I saw this form, I then said to myself, if we restrict the domain (the choice of elements of G) to be strictly positive ,then, if I am not mistaken, the objective function is convex.
What do you think?