Clear[U, V, 
      T]
U = {u1, u2, 
          u3};
V = {v1, v2, 
          v3};
T = {t1[u1, u2, u3], t2[u1, u2, u3], t3[u1, u2, u3]};

Fhat := 
        Transpose[{(∂_u1) T, (∂_u2) T, (∂_u3) 
              T}];
MatrixForm[
      Fhat]
Eigensystem[Fhat]

Eigensystem[(({{0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}}))]

F = {v1, v2, v3, 
          t1[u1, u2, u3], t2[u1, u2, u3], 
          t3[u1, u2, u3]};
FluxMatrix = 
        Transpose[{(∂_u1) F, (∂_u2) F, (∂_u3) 
              F, (∂_v1) F, (∂_v2) F, (∂_v3) 
              F}];
MatrixForm[
      FluxMatrix]
{EVal1, EVal2, EVal3, EVal4, 
        EVal5, EVal6} = 
      Eigenvalues[FluxMatrix]
{EVec1, EVec2, 
        EVec3, EVec4, EVec5, EVec6} = Eigenvectors[FluxMatrix]