(* Following Section 3.12 in  "The fuzzy space construction kit" by Andreas Sykora, Calculating the index for r=1 *)

r=1;

n=6;

V ={{8/10,1,1,0,0,0},{0,0,0 ,1,0,0},{0,0,16/10,r,1,0},{0,0,0,8/10,0,1},{0,0,0,0,24/10,1},{0,0,0,0,0,16/10}};

MatrixForm[V]

Z = (1/10)*{{0,0,0,0,0,0},{0,13,0,0,0,0},{0,0,13,0,0,0},{0,0,0,26,0,0},{0,0,0,0,26,0},{0,0,0,0,0,39}};

MatrixForm[Z]

X = (1/2)*ConjugateTranspose[V]+(1/2)*V;

MatrixForm[X]

Y = (I/2)*ConjugateTranspose[V]-(I/2)*V;

0

MatrixForm[Y]

sigma1 = {{0,1}, {1,0}};

sigma2 = {{0,-I},{I,0}};

sigma3 = {{1,0},{0,-1}};

loclzr = KroneckerProduct[sigma1, X- x*IdentityMatrix[n]] + 
KroneckerProduct[sigma2,Y-y*IdentityMatrix[n]]+ 
KroneckerProduct[sigma3, Z-z*IdentityMatrix[n]] ;

charpoly = Det[loclzr];

ContourPlot3D[charpoly== 0, {x,-1,3}, {y,-1, 1}, {z, 0,4},Contours->{{1,LightBlue}},PlotPoints->100,ViewPoint->{2,-18,2}]

Eigenvalues[N[ReplaceAll[loclzr,{x->2,y->0,z->1/4}]]](*This point has index -1 *)