A = {{a11, a12}, {a21, a22}};
l = {{l1, l2}};
F = {{F1, F2}};
P = {{P1, P2}};
X = DiagonalMatrix[{x1, x2}];
Solve[F == P.Inverse[X].A.X + w*l, {x1, x2}]

{}

Solve[F == P.Inverse[X].A.X + w*l, {P1, P2}]

{{P1 -> -((-a21 F2 x1 + a21 l2 w x1 + a22 F1 x2 - a22 l1 w x2)/((a12 a21 - a11 a22) x2)), P2 -> (-a11 F2 x1 + a11 l2 w x1 + a12 F1 x2 - a12 l1 w x2)/((a12 a21 - a11 a22) x1)}}