# Expectation of the sum of squares of a covariant normal samples

Given a series of samples $y_1, y_2, \ldots, y_n \sim N(0,\Sigma)$, I'm looking to find the expection, $E(y^T y)$. It's fairly easy to show that $y^T \Sigma^{-1} y \sim \chi^2$, but then I get stuck. Maybe there isn't a closed form solution for $E(y^T y)$?

Thanks

Edited: Removed expectation around product.

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$E(y^T y) = tr(\Sigma)$
Also, I think you mean to say that $y^T \Sigma^{-1} y \sim \chi ^2$, not $E(y^T \Sigma^{-1} y) \sim \chi ^2$