I am trying to find out, if there is a formula for finding the inverse of the sum of two orthogonal projections. So basically my questions is:
If $\left[\mathbf{A},\mathbf{B}\right]$ is full rank, then $\left(\mathbf{A}\left(\mathbf{A}^{\mathrm{H}}\mathbf{A}\right)^{-1}\mathbf{A}^{\mathrm{H}} + \mathbf{B}\left(\mathbf{B}^{\mathrm{H}}\mathbf{B}\right)^{-1}\mathbf{B}^{\mathrm{H}}\right)^{-1}$ = ?