I am trying to find the sum of two quadratic forms, $(x-y)^TA^{-1}(x-y) + (y-z)^TB^{-1}(y-z)$, where $A$ and $B$ are symmetric positive $p\times p$ definite matrices. $x,y,z$ here are $p\times 1$ vectors.
Specifically, I would like to get it into the form:
$$ (x-y)^TA^{-1}(x-y) + (y-z)^TB^{-1}(y-z) = v^TCv $$
In other words, I would like to combine them into another quadratic matrix with vector $v$ and positive semi-definite matrix $C$. Would anyone know how to do this? I have tried expanding and it is a mess and I am positive that I need to introduce additional variables by using the identity matrix, i.e., $BB^{-1} = I$ and multiplying this to strategic elements in my matrix.
This problem arises in the application of Bayes Theorem to a multivariate normal with a multivariate normal likelihood. Derivations can be found in many textbooks that discuss Bayesian statistics and applications to inverse problems. See for example Tarantola's "Inverse Problem Theory and Methods for Model Parameter Estimation."