How to find a transformation matrix between two set of operators?

The following set of operators $ P_1,P_2,P_3,P_4$ and $ Q_1,Q_2,Q_3,Q_4$ are related as $ P_i=\sum_i w_{ij} Q_j$ . How can I find the matrix $ W=[w_{ij}]$ ?

 Q1 = {{0, 0}, {Sqrt[x*y], 0}} Q2 = {{0, Sqrt[x*(1 - y)]}, {0, 0}} Q3 = Sqrt[y]*{{Sqrt[1 - x], 0}, {0, 1}} Q4 = Sqrt[1 - y]*{{1, 0}, {0, Sqrt[1 - x]}} P1 = {{0, 0}, {Sqrt[x*y], 0}} P2 = {{0, Sqrt[x*(1 - y)]}, {0, 0}} P3 = (Sqrt[2 - x - Sqrt[4 - 4*x + x^2 - 4*x^2*y + 4*x^2*y^2]]/         (Sqrt[2]*Sqrt[1 + (1/4)*                  ((-x + 2*x*y +                Sqrt[4 - 4*x + x^2 - 4*x^2*y + 4*x^2*y^2])/                       Sqrt[1 - x])^2]))*      {{-((-x + 2*x*y + Sqrt[4 - 4*x + x^2 - 4*x^2*y + 4*x^2*y^2])/               (2*Sqrt[1 - x])), 0}, {0, 1}} P4 =        (Sqrt[2 - x + Sqrt[4 - 4*x + x^2 - 4*x^2*y + 4*x^2*y^2]]/           (Sqrt[2]*Sqrt[1 + (1/4)*                    ((-x + 2*x*y -                 Sqrt[4 - 4*x + x^2 - 4*x^2*y + 4*x^2*y^2])/                         Sqrt[1 - x])^2]))*        {{-((-x + 2*x*y - Sqrt[4 - 4*x + x^2 - 4*x^2*y + 4*x^2*y^2])/                 (2*Sqrt[1 - x])), 0}, {0, 1}}