In Statistics, we often talk about diagonalising the Covariance matrix (like in Principal Analysis Components) but what would be the interest and the usefulness of diagonalising the Fisher matrix : How could I exploit this diagonal matrix ?
Initially, I thought that it would allow me to do cross-correlations between 2 Fisher matrix by summing the 2 diagonalised matrix and doing the sum to get an estimator $\sigma_{\hat{\tau}}$ on each parameter :
$$\dfrac{1}{\sigma_{\hat{\tau}}^{2}}=\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}$$
But it doesn't give the expected gain for the constraints on parameters.
So, I wonder if there are other utilities to perform these diagonalisations on Fisher matrix (with a preference for carrying out cross-correlations if possible).
Any help/suggestion/clue/remark is welcome.