SetDirectory["/www/user/fdahl/papers/Conjugation/"];
<<kappaLib.m
<<Petrov.m

Class XVII: (211 1bar(1))

B=(({{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}}));

V=(({{ lam1, 1, 0, 0, 0, 0}, {0, lam1, 0, 0, 0, 0}, {0, 0, lam2, 0, 0, 0}, {0, 0, 0, lam3, 0, 0}, {0, 0, 0, 0,  sigma1, tau1}, {0, 0, 0, 0, -tau1,  sigma1}}));

Eigenvalues[V]

W=(({{0, eps1, 0, 0, 0, 0}, {eps1, 0, 0, 0, 0, 0}, {0, 0, eps2, 0, 0, 0}, {0, 0, 0, eps3, 0, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0}}));

Eigenvalues[W]

eps2 and eps3 have the same block size so we may assume that eps2 <= esp3. From Eigenvalues[W] we see that eps2 and eps3 can not be of the same sign.

W=W/.{eps2->-1, eps3->1};
Eigenvalues[W]

eps1 is arbitrary

Sort[Eigenvalues[W]/.{eps1->-1}]
Sort[Eigenvalues[W]/.{eps1->1}]

S=(({{0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, (1/(Sqrt[2])), (1/(Sqrt[2])), 0, 0}, {eps1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, -(1/(Sqrt[2])), (1/(Sqrt[2])), 0, 0}}));

Check that S is in the set mathcal(S)

Transpose[S].B.S==W

Compute result

res=S.V.Inverse[S];
res//MatrixForm

Petrov[res]

Export notebook as .pdf

NotebookPrint[SelectedNotebook[],"/www/user/fdahl/papers/Conjugation/notebooks/ClassXVII.pdf"]