On the Positivity of a Class of Cauchy-Like Matrices

Let $0<{\lambda }_1<\cdots <{\lambda }_n$. Motivated by a problem related to Lyapunov equations we consider a class of Cauchy-like matrices whose elements have the form $C_{ij}=\frac{r_i(k,l)+r_j(k,l)}{{\lambda }_i+{\lambda }_j},$ where for any pair $1\le k,l\le n$, $r_i(k,l)$ are functions of $\{{\lambda }_1,\cdots ,{\lambda }_n\}\setminus {\lambda }_i$. We show that these matrices are positive semidefinite for every pair $1\le k,l\le n$. After passing to the reciprocal variables $x_i=1\mathrm{/}{\lambda }_i$, the problem is reduced by a diagonal congruence to the positivity of a two-parameter family $A_n^{(p,q)}(x)$. The proof introduces a singular augmented matrix ${\mathcal{H}}_n^{(p,q)}(x)$, proves its singularity by Cauchy-kernel generating function identities, and then proves positive semidefiniteness by induction on $n$ using the principal-minor criterion.