I am working on a problem involving the the polarized Hessian covariant in Cartesian coordinates on $\mathbb{R}^2$
$[a,b] = \frac{1}{2} \frac{\partial ^2 a}{\partial x ^2} \frac{\partial ^2 b}{\partial y ^2} + \frac{1}{2} \frac{\partial ^2 a}{\partial y ^2} \frac{\partial ^2 b}{\partial x ^2} - \frac{\partial ^2 a}{\partial x \partial y} \frac{\partial ^2 b}{\partial x \partial y}$
and I am also encountering a similar-looking operator acting on a sole symmetric matrix
$[Q] = \frac{\partial ^2 Q_{11}}{\partial x ^2} + \frac{\partial ^2 Q_{22}}{\partial y ^2} - 2 \frac{\partial ^2 Q_{12}}{\partial x \partial y}$
Because this operator "looks similar" to the polarized Hessian and is appearing in a similar context I am trying to establish a connection between the two. In particular since Q is symmetric I can treat it as a bilinear form, so I can write this $[Q]$ in terms of the polarized Hessian as
$[Q] = [Hess(Q),S] = [(\frac{\partial ^2 Q_{11}}{\partial x ^2}x^2 + \frac{\partial ^2 Q_{22}}{\partial y ^2}y^2 + \frac{\partial ^2 Q_{12}}{\partial x \partial y}xy), (\frac{1}{4}x^2 + \frac{1}{4}y^2 + 2xy)]$
where I am treating $Hess(Q)$ as constant coefficients for the first form and where $S = \left( \begin{array}{ccc} \frac{1}{4} & 2 \\ 2 & \frac{1}{4} \end{array} \right) $ is the symmetric matrix corresponding to the second form in the above bracket.
My question is, does this matrix S have any particular significance? I chose it only to write $[Q]$ in terms of the Hessian/polarized Hessian, so it seems arbitrary. Alternatively, does anybody recognize this operator $[Q]$? I have very little background in invariant/covariant theory.