Problem with Covariance matrix using diagonal loading involved in calculation of eigenvalues
You can write 

$$ R=YY^H $$
where $Y$ is a matrix of size $N\times N_f$ and $N$ is the dimension of $y_k$. $Y$ contains all the measured $y_k$ as its columns. 

Then, the rank of $R$ is upper bounded by $N_f$. In particular, if $N_f&lt;N$, $R$ will always be a singular matrix. So, if you have too few measurements, you will likely run into the problem you described.

For your second part, how do the Eigenvalues of $R$ and $R+\sigma I$ relate:
Consider the eigenvalue decomposition of $R$ by
$$ R=PVP^{-1}$$
where $P$ contains the eigenvectors and V is the diagonal matrix with the eigenvalues (they can be zero). Then you have

$$ R+\sigma I=P V P^{-1} + \sigma I = P(V+\sigma I)P^{-1} $$

This means that the eigenvalues are all increased by $\sigma$.