# Is there any connection between this matrices

Matrices I discuss are all $N\times N$ hermitian matrices. Define two positive (semi)definite matrices $H_1$ and $H_2$. Define the following matrices \begin{align} P_1&=H_1+(I+H_2)^{-1} \\ P_2&=(I+H_1)^{-1}+H_2 \end{align} I was just curious if there are any connections between them. It can be from any perspective, eigenvalues, rank, eigenbasis, simultaneous diagonalization or any such concept. Even special cases are welcome, for instance, say they are rank-one matrices, Does it make any difference?

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This is not an answer but I wanted to show some graphs so I am posting it. I generated a 1000 FULL RANK Positive Definite matrices $H_1$ and $H_2$ and ran some preliminary simulations. At least for full rank matrices, the spectrum doesn't seem to tell anything interesting. This seems like a really interesting question and I would really appreciate any comments on my attempt of using simulation to try answering this question. (The results were size invariant)..