= Modified trace form on real matrices
{title2=$\phi(A,B)=\operatorname{tr}(AB)-\operatorname{tr}(A)\operatorname{tr}(B)$}
On real $n$ by $n$ matrices with $n\geq2$, split into traceless symmetric matrices, skew-symmetric matrices and scalar matrices. These summands are orthogonal for the displayed <symmetric bilinear form>. The form is positive on the first summand, negative on the second and negative on the scalar identity direction. Its <inertia of a bilinear form> is $(n(n+1)/2-1,n(n-1)/2+1,0)$, its rank is $n^2$ and its <signature> is $n-2$. Indeed $\operatorname{tr}(S^2)=\|S\|_F^2$, $\operatorname{tr}(K^2)=-\|K\|_F^2$ and $\phi(tI,tI)=-n(n-1)t^2$.
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