Pensieve header: MS, manually compute matrix signature.

A=IdentityMatrix[2]

A1=A[[2;;,2;;]]

A1[[2;;,2;;]]

Outer[Times,{a,b,c},{1,2,3}]

MS[{}]=0;
MS[A_?MatrixQ]/;A==A\[Transpose]:=Module[{k,a,A1,j,l},
{k}=FirstPosition[A[[1]],x_/;x!=0,{None}];
Switch[k,
None, MS[A[[2;;,2;;]]],
1,(
a=A[[1,1]];
Sign[a]+MS[
A[[2;;,2;;]]-Outer[Times,A[[2;;,1]],A[[1,2;;]]]/a
]),
_,(
A1=A; A1[[k]]/=A[[k,1]]; A1[[All,k]]/=A[[1,k]];
a=A1[[k,k]]; A1[[k]]-=a A1[[1]]/2; A1[[All,k]]-=a A1[[All,1]]/2;
For[j=2,j<=Length@A,++j,If[j!=k,
A1[[j]]-=A1[[j,1]]A1[[k]]+A1[[j,k]]A1[[1]];
A1[[;;,j]]-=A1[[1,j]]A1[[;;,k]]+A1[[k,j]]A1[[;;,1]]
]];
l=Complement[Range@Length@A,{1,k}];
MS[A1[[l,l]]]
)
]
]

A=Table[RandomInteger[{-1,1}],7,7];
A=A+A\[Transpose];
{MatrixSignature[A],MS[A]}

MS[(({{0, 4, 0, 6, 5}, {4, 4, -1, 6, 3}, {0, -1, -2, -3, -2}, {6, 6, -3, 2, 1}, {5, 3, -2, 1, 0}}))]