After writing up some math, I ended up with a term like so:
$\left(A^{-1} + B^{-1}\right)^{-1}$
where $A$ and $B$ are 2 covariance matrices.
1) Can I be sure that this expression is meaningful? (i.e. $\left(A^{-1} + B^{-1}\right)$ is not singular)
2) If $A$ and $B$ were numbers, I could rewrite this as $\frac{AB}{A+B}$. Is there a similar formula valid for matrices? Ultimately I wonder if this expression can be simplified:
$z^T\left(A^{-1} + B^{-1}\right)^{-1}z$
where
$z=A^{-1}x + B^{-1}y$ , for some vectors $x$ and $y$.
If $A$ and $B$ are invertible covariance matrices, they must be positive definite; hence their inverses must be positive definite. But the sum of two positive definite matrices is again positive definite. (Proof: when $u\neq0$, $u^T(A+B)u=u^tAu+u^tBu>0+0=0$). Therefore $A^{-1}+B^{-1}$ is positive definite, hence invertible.
See Inverse of the sum of matrices. Note your formula has at least four different interpretations for matrices: $AB(A+B)^{-1}$, $(A+B)^{-1}AB$, etc.