# Non-negative Quadratic forms with Exterior Forms

Hello All,

I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.

Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over $\mathbb{R}$ and let $T:\Lambda^{2}(X)\rightarrow \Lambda^{2}(X)$ be a symmetric linear operator satisfying $$\langle T(\omega),\omega\rangle\geqslant 0,\text{ for all }\omega\in \Lambda^{2}(X)\text{ with }\omega\wedge\omega=0.$$ Is it true that, for some $A\in \Lambda^{4}(X)$, $$\langle T(\omega),\omega\rangle\geqslant 0,\text{ for all }\omega\in \Lambda^{2}(X)\text{ with }\langle A;\omega\wedge\omega\rangle=0?$$

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It may have some some connection with the following thread: mathoverflow.net/questions/118037/… –  tatin Feb 4 '13 at 14:09