Prove that $$\left( A+BB^T \right)^{-1} B = A^{-1}B \left( I+B^TA^{-1}B \right)^{-1}$$ where $A$ is an $n \times n$ matrix and $B$ is an $n \times k$ matrix. Assume that all inverses exist.
I used the Sherman-Morrison identity , but I didn't get the exact form of this identity. Please help!
$B(I+B^{T}A^{-1}B)=B+BB^{T}A^{-1}B=(A+ BB^{T}) (A^{-1}B)$. Now multiply on the left by $(A+BB^{T})^{-1}$ an don the right by $(I+B^{T}A^{-1}B)^{-1}$.