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.