I want to know if there is a way to simplify, or a closed form solution of $tr(\Sigma^{-1})$ where $\Sigma$ is a symmetric positive definite matrix.
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Let $A$ be symmetric positive definite matrix hence $\exists$ a diagonal matrix $D$ whose diagonal entries are nonzero and $A=P D P^{-1}$ so $A^{-1} = P D^{-1} P^{-1}$ and $Tr(A^{-1})= Tr(D^{-1})$. Now $D$ being diagonal matrix with non zero diagonal entries $D^{-1}$ has diagonal entries reciprocal of the diagonal entries of $D$ so $Tr(D^{-1})$ is sum of the inverses of the diagonal entries of $D$.
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9$\begingroup$ You might want to mention that the values in $D$ are the Eigenvalues (usually denoted $\lambda_j$) of $A$ and can be obtained from $\det(A-I\lambda_j)=0$ (where $I$ denotes the fitting identity matrix) $\endgroup$ Aug 13, 2014 at 9:50