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

Class XXII: (31 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, 1, 0, 0, 0}, {0, 0, lam1, 0, 0, 0}, {0, 0, 0, lam2, 0, 0}, {0, 0, 0, 0,  sigma1, tau1}, {0, 0, 0, 0, -tau1,  sigma1}}));

Eigenvalues[V]

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

Possible choices for eps_i:

Eigenvalues[W]

s[i_]:=(-1)^i

For[i1=0,i1≤1,i1++,
For[i2=0,i2≤1,i2++,
Print[ToString[{s[i1],s[i2]}]];
Print[Sort[Eigenvalues[W]/.{eps1->s[i1],eps2->s[i2]}]];
]
]

4 sign possibilities: (e1,e2)= {+-, -+}

W=W/.{eps2->-eps1};
Eigenvalues[W]

(* See ClassXXII_Solve.nb *)
S=(({{0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, {0, (1/(Sqrt[2])), 0, (1/(Sqrt[2])), 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, eps1, 0, 0, 0}, {0, (eps1/(Sqrt[2])), 0, -(eps1/(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];
r=Simplify[res];
r//MatrixForm

Petrov[r]

Export notebook as .pdf

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