# Interpretation of (diagonalized) inverse covariance matrix

There are several threads here about covariance matrix and inverse covariance matrix interpretation (here, here or here).

However, I was wondering how to interpret the inverse covariance matrix (or precision matrix) as the covariance matrix can be diagonalized (since it's a definite positive matrix). $$\Sigma = P D P^{-1}$$ ($\Sigma$ being the covariance matrix and $D$ is a diagonal matrix). Then $$\Sigma^{-1} = P D^{-1} P^{-1}$$

How to interpret this diagonal precision matrix $D^{-1}$ ? I think I don't get exactly the signification of the eigenvectors $P$ .

• The eigenvectors are the same in both cases--$P$ and $P^{-1}$ play their roles in both formulas--and therefore are subject to exactly the same interpretations. Which part don't you "get exactly"? – whuber Jun 8 '15 at 21:53