begin{align}label{gs_diff_homog_3}
begin{bmatrix} hat a(omega, x_3) \ hat b(omega,x_3) end{bmatrix} &= underbrace{begin{bmatrix} chi_1(omega,x_3) & chi_2(omega,x_3) \ chi_3(omega,x_3) & chi_4(omega,x_3) end{bmatrix} begin{bmatrix} chi_1(omega,0) & chi_2(omega,0) \ chi_3(omega,0) & chi_4(omega,0) end{bmatrix}^{-1}}_textrm{$mathbf{P_{omega}}(0,x_3)$} \ &timesunderbrace{begin{bmatrix} chi_1(omega,0) & chi_2(omega,0) \ chi_3(omega,0) & chi_4(omega,0) end{bmatrix} begin{bmatrix} c_1 \ c_2 end{bmatrix}}_textrm{$[hat a(omega,0),hat b(omega,0)]^{T}$}. nonumber
end{align}