I have two square invertible positive definite symmetric matrices, $A_1$ and $A_2$, and there is a third square invertible positive definite symmetric matrix which I do not know the value of, $B$. All matrices are of equal size.
I know my matrices satisfy the below equation;
$(A_1+B)^{-1}+(A_2+B)^{-1}=\alpha B$
For some scalar $\alpha$ I know the value of. Is there any way I can solve this expression for $B$? Preferably I'd like to solve this theoretically but if anyone knows of a computational way I am open to suggestions too.
Not a solution, but an approach. Multiplying the equation (from the left side) with either $(A_1+B)$ and $(A_2+B)$ yields $$(A_2+B)+(A_1+B) = \alpha B^3 + \alpha (A_1+A_2)B^2 + \alpha A_1A_2 B\, , $$ which is a polynomial in $B$ of degree $3$. Since this solution would need to satisfy the case of $1\times 1$ matrices, we firstly try to solve the associtated scalar polynomial equation for $B$. These solutions then would be a candidate for the matrix equation, if you can make sense out of roots of matrices.