The Kulkarni–Nomizu product of two symmetric matrices $h_{ij}$ and $k_{kl}$ is $$ (h\wedge \!\!\!\!\!\!\bigcirc k)_{ijkl} = h_{ik}k_{jl} + h_{jl}k_{ik} - h_{il}k_{jk} - h_{jk}k_{il}. $$ 'Squaring' a matrix with this product, we get \begin{align} (h\wedge \!\!\!\!\!\!\bigcirc h)_{ijkl} &= h_{ik}h_{jl} + h_{jl}h_{ik} - h_{il}h_{jk} - h_{jk}h_{il} \\ &= 2(h_{ik}h_{jl} - h_{il}h_{jk}), \end{align} (which I recognize as twice the determinant of $h_{ij}$ in 2D). In particular, if $h_{ij}$ is the Hessian of a scalar function $f$, $$ h_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j} \equiv f_{,ij}, $$ then half the product is $$ \tfrac{1}{2}(h\wedge \!\!\!\!\!\!\bigcirc h)_{ijkl} = f_{,ik}f_{,jl} - f_{,il}f_{,jk}. $$ Again, in 2D, this is the determinant of the Hessian, which I understand to be the discriminant at a critical point (only in 2D).
Finally, if we trace this over the indices $(i,k)$, and again over $(j,l)$, we get \begin{align} \sum_i \sum_j \tfrac{1}{2}(h\wedge \!\!\!\!\!\!\bigcirc h)_{ijij} &= \sum_i h_{ii} \sum_j h_{jj} - \sum_i \sum_j h_{ij} h_{ji} \\ &= (h_{11}+h_{22}+\ldots)^2 - (h_{11}^2+h_{22}^2+\ldots+h_{12}h_{21}+\ldots) \\ &= 2(h_{11}h_{22}-h_{21}h_{21})+2(h_{11}h_{33}-h_{31}h_{31})+\ldots \\ &= 2\sum (\text{determinants of $2\times2$ minors of $h$.}) \\ &= 2\sum (\text{products of eigenvalues of $2\times2$ minors of $h$?}) \end{align} If $h$ is the Hessian of $f$, this is $$ 2(f_{,11}f_{,22}-f_{,21}f_{,21})+2(f_{,11}f_{,33}-f_{,31}f_{,31})+\ldots $$ or (I think) $$ 2(\lambda_1\lambda_2 + \lambda_1\lambda_3 + \ldots + \lambda_2\lambda_3 + \ldots) $$
Questions
- What would possess me to take the Kulkarni–Nomizu product of a Hessian? Has anyone seen anything like this?
- What does the product tell us about $f$ geometrically? (Not necessarily at a critical point).
- What does the double trace tell us about $f$?
- What does the sum of pairs of eigenvalues mean?? EDIT: I just found out this is called the 2nd symmetric function of eigenvalues. What does it mean?