Is the sum of two singular covariance matrices also singular? I have two sample covariance matrices, computed from $n$ samples, less than $p$ variables: they are singular then.
I know that the sum of two covariance matrices is also a covariance matrix.
My question is, if both are singular, is this sum also singular?
And if both are not singular, is this sum also not singular?

No, consider the singular covariance matrix $\pmatrix{1 & 0\\ 0 & 0}$ summed with the singular covariance matrix $\pmatrix{0&0\\0&1}$.
Regarding your second question, the sum of two nonsingular covariance matrices is also nonsingular. This is because the set of nonsingular covariance matrices is exactly the set of positive definite matrices (which is closed under positive combination, see here for the sum case).
