TITLE: Trace of an Inverse Matrix
QUESTION [23 upvotes]: 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.

REPLY [36 votes]: 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 $D$.