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 $P$ .

Thanks for your help.

• 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
• What I don't get is the signification of a covariance matrix when this matrix is diagonal. And also the signification of the inverse covariance matrix when this matrix is diagonal. – max Jun 9 '15 at 16:46
• I think your last comment clarified what you were asking; this was not very clear in your original question. Please see my answer below. – usεr11852 Oct 26 '15 at 7:42