If I understand your question correctly, there is no unique solution. You can understand this by looking at a 2-by-2 system.
P = [a b;c d]
V = [e;f]
Vs = [g;h]
Now from P*V = Vs, we know that:
a*e + b*f = g
c*e + d*f = h
Thus if V and Vs are known, then e, f, g, and h are known. But we still have two equations and four unknowns ( a , b, c, and d ). So unless there are two more ways of limiting the choices for the four unknowns, we are stuck.
For a larger system, the problem only gets worse.
