I have the following system of differential equations:

$ \begin{pmatrix}\dfrac{d\eta}{dt} \\\dfrac{d\eta^{\dagger}}{dt} \end{pmatrix} = \begin{pmatrix}(1+I\kappa)(w-IJ) & R_1+i R_1\kappa \\-R_1+IR_1\kappa & (IJ+w)(-1+i\kappa)\\\end{pmatrix} \begin{pmatrix}\eta \\\eta^{\dagger}\end{pmatrix}$

For solve this system, I diagonalize the intermediate matrix and write the solution in terms of the normal modes. On Mathematica:

s = {{(1 + I \[Kappa]) (w - I J), R1 + I R1 \[Kappa]}, {-R1 +  I R1 \[Kappa], -(1 - I \[Kappa]) (w + I J)}};
{vals, vecs} = FullSimplify[Eigensystem[s ]];
NormVector = Transpose[Normalize /@ vecs];
NormalModes = {{C1*Exp[I*vals[[1]]*t]}, {C1*Exp[I*vals[[2]]*t]}};
Solution = NormVector.NormalModes /. {w -> 0.27, \[Kappa] -> 0.01, R1 -> 0.025,
J -> 0.005}

In this case Solution = $ (-0.0463375-0.000463375 i) \text{C1} e^{(0.0023\, -0.26889 i) t}-(0.998876\, +0.00998876 i) \text{C1} e^{(0.0023\, +0.26889 i) t}\\ $ $ 0.998926 \text{C1} e^{(0.0023\, -0.26889 i) t}+0.0463398 \text{C1} e^{(0.0023\, +0.26889 i) t} \ $ For the other hand, I define the constants before to do the calculation, in this way,

\[Kappa] = 0.01
R1 = 0.025;
J = 0.005;
w = 0.27;
  s = {{(1 + I \[Kappa]) (w - I J), R1 + I R1 \[Kappa]}, {-R1 +  I R1 \[Kappa], -(1 - I \[Kappa]) (w + I J)}};
{vals, vecs} = FullSimplify[Eigensystem[s ]];
NormVector = Transpose[Normalize /@ vecs];
NormalModes = {{C1*Exp[I*vals[[1]]*t]}, {C1*Exp[I*vals[[2]]*t]}};
Solution2 = NormVector.NormalModes /. {w -> 0.27, \[Kappa] -> 0.01, R1 -> 0.025,
J -> 0.005}

In this case $ Solution2 = (0.998926\, +0. i) \text{C1} e^{(0.0023\, +0.26889 i) t}-(0.0463375\, +0.000463375 i) \text{C1} e^{(0.0023\, -0.26889 i) t} \\$

$ (0.998926\, +0. i) \text{C1} e^{(0.0023\, -0.26889 i) t}+(-0.0463375+0.000463375 i) \text{C1} e^{(0.0023\, +0.26889 i) t} $

That is the correct result, I do not understand why I obtain different results. In the first solution, the imaginary part of the second row of the matrix is missing.

Thanks