https://www.cs.nyu.edu/~roweis/notes/gaussid.pdf contains expressions (p.2, 6e, 6f) for the normalization constant for the product of two multivariate Gaussian pdfs, with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively. The product has mean $c$ and covariance matrix $C$. To simplify the notation, I'm writing here only the exponent for the normalization factor, and ignore the -1/2:
$a^TA^{-1}a+b^TB^{-1}b-c^TC^{-1}c$
Another source, http://www.gaussianprocess.org/gpml/chapters/RWA.pdf, gives the following equation for the same: (page 200, A.8):
$(a-b)^T(A+B)^{-1}(a-b)$
After a number of failed attempts to derive the latter from the former, I think I'm missing a point somewhere.
Is there a straightforward solution? Any hint that could help? Or any source available on the web with the derivation?
One way is to first expand the right summand in the first equation - i.e. perform the vector-matrix multiplications.
Then modify the result using for example the identity given here: Inverse of a sum of symmetric matrices.
Further modify using the matrix inversion theorem (here a special case: $U=V=I$) - it's actually shown on the next page of one of the sources given in the question:http://www.gaussianprocess.org/gpml/chapters/RWA.pdf, p.201 1st paragraph)
What one is left with after further rearranging and simplifying is the second equation plus the two left summands of the first equation - so plugging this result into the first equation and performing the subtractions gives the second equation (times -1, which had been ignored here).