Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as \begin{align} \mathbf{C}=\left(\mathbf{I}*\frac{1}{\alpha}+\mathbf{B}\right)^{-\frac{1}{2}}\mathbf{A}\left(\mathbf{I}*\frac{1}{\alpha}+\mathbf{B}\right)^{-\frac{1}{2}} \end{align} where $\mathbf{I}$ is the $N\times N$ identity matrix, $\alpha$ is a positive variable. The question is to find the $\alpha$ which maximizes the largest eigenvalue of $\mathbf{C}$.

**Background**: This comes from a real world wireless problem where the largest eigenvalue represents the best power allocation at the transmitter. $\mathbf{A}$ and $\mathbf{B}$ represent the channel matrices.