The spectral map for weighted Cauchy matrices is an involution

Let $N$ be a natural number. We consider weighted Cauchy matrices of the form \[ \mathcal{C}_{a,A}=\left\{\frac{\sqrt{A_j A_k}}{a_k+a_j}\right\}_{j,k=1}^N, \] where $A_1,\dots,A_N$ are positive real numbers and $a_1,\dots,a_N$ are distinct positive real numbers, listed in increasing order. Let $b_1,\dots,b_N$ be the eigenvalues of $\mathcal{C}_{a,A}$, listed in increasing order. Let $B_k$ be positive real numbers such that $\sqrt{B_k}$ is the Euclidean norm of the orthogonal projection of the vector \[ v_A=(\sqrt{A_1},\dots,\sqrt{A_N}) \] onto the $k$'th eigenspace of $\mathcal{C}_{a,A}$. We prove that the spectral map $(a,A)\mapsto (b,B)$ is an involution and discuss simple properties of this map.