Let $\Sigma_1$ and $\Sigma_2$ be two known symmetric and positive definite matrices such that they can be decomposed as $$ \Sigma_1 = A + B_1 \;\;\;\; \Sigma_2 = A+B_2 $$ where A, $B_1$ and $B_2$ are symmetric matrices. We want to find the solutions for $A$, $B_1$ and $B_2$ such that this solution is unique. In particular, by solving these two matrix equations you can have many possible solutions bringing to the same $\Sigma_1$ and $\Sigma_2$, for example
1) $A = a$, $B_1=b_1$ and $B_2 =b_2 $
2) $A = a/2$, $B_1 = a/2 +b_1$ and $B_2 = a/2 + b_2$.
How can I have a unique solution? My intuition is to impose the constraint that the intersection of the span of $B_1$ and $B_2$ to be empty. Do you have a proof for that or other constraints?