How to calculate the Diagonal loading factor evaluate calculate the inversion of a covariance matrix
Short answer: just use $\sigma = 10^{-8}$.
Covariance matrices have eigenvalues $\geq 0$ (theoretically),
so $Ri + 10^{-8} \, I$ will have eigenvalues $\geq 10^{-8}$,
safely non-singular.

A longer answer:
split Covar = S + N, "signal" + "noise", by eigenvalues or by SVD,
Singular-value decomposition
aka PCA, Principal component analysis.
This has several advantages:


gives you an idea of how noisy your Covariance is -- a good thing to know
you can choose a cutoff, e.g. 98 % of the sum of the eigenvalues,
or a number of terms, e.g. 3, to plot and look at
fast, rock-solid implementations:
in Python,
numpy eigvalsh,
numpy svd
and sparse
scipy svds 
(sparse, dense or LinOp).


Be careful with large covariance matrices:
eigenvalues can be &lt; 0 because of roundoff,
and they can be very noisy -- see
here
on stats.stack .

See Press et al. for a good discussion of SVD:
"Solution by use of Singular-value decomposition",
Numerical Recipes p. 793 ff.