How to analyse the relationship between the expression and matrix eigenvalues

Question

I have an expression: $$2\cdot\mathbb{R}\left ( \frac{P_{a}^{H}\left ( A^{H}A \right )P_{a}}{P^{H}\left ( A^{H}A \right )P}+\frac{\left | P_{a}^{H}\left ( A^{H}A \right )P \right |^{2}}{\left | P^{H}\left ( A^{H}A \right )P \right |^{2}} \right)$$ where $P$ is a $2\times 1$ matrix with parameters of $\alpha, P_{a}$ means $\frac{\partial P}{\partial \alpha },$ $H$ means Conjugate transpose, $A$ is a $25\times 2$ matrix. The question is: I know the two eigenvalues of $\left ( A^{H}A \right ), \lambda _{1}$ and $\lambda _{2},$ how to analyse the relationship between $\frac{\lambda _{1}}{\lambda _{2}}$ and the expression above, it can be a size relationship or other?