On the Positivity of a Class of Cauchy-Like Matrices

Abstract

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 $x_i=1/\lambda_i$, 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.