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Here is the question

Find positive and negative indexes of inertia for the function $q(x)=TrX^2$ on the space $M_n(R)$

I did some work, first I suppose $X$ as a n by n matrix, then $TrX^2=a_{11}^2 +...+a_{nn}^2+2(a_{ij}a_{ji})$

It seems like that all terms are positive, unless $a_{ji}=-a_{ji}$, hence the positive index will be $3n$ and the negative index is $0$.

Am I right?

Thanks.