On a question of Bhatia, Friedland and Jain II (Q6593275)

Let \(p_1<\ldots<p_n\) be positive real numbers. The Loewner matrix corresponding to the function \(f(t)=t^r\), \(r\ge 0\), is \(L_r=(l_{ij}^{(r)})\) with\N\[\Nl_{ij}^{(r)}=\frac{p_i^r-p_j^r}{p_i-p_j} \ \textrm{for} \ i\ne j,\quad l_{ij}^{(r)}=rp_i^{r-1} \ \textrm{for} \ i=j.\N\]\NConsider the Cauchy matrix corresponding to the \(p_i\)s. The Hadamard inverse of its \(r\)-th Hadamard power, \(r>0\), is \(P_r=(p_{ij}^{(r)})\) with\N\[\Np_{ij}^{(r)}=(p_i+p_j)^r.\N\]\N\textit{R. Bhatia} and \textit{T. Jain} [J. Spectral Theory 5, 71--87 (2015; Zbl 1321.15017)] determined the inertia of \(P_r\). They and \textit{S. Friedland} [Indiana Univ. Math. J. 65, 1251--1261 (2016; Zbl 1354.15005)] determined that of \(L_r\). The inertias of \(L_{r+1}\) and \(P_r\) turned out to be the same, which motivated to study relations between these matrices. The present authors show that if \(L_{r+1}\) and \(P_r\) are nonsingular, then there is a real nonsingular upper triangular matrix \(X\) such that \(X^TP_rX=L_{r+1}\). They also show that if \(L_{r+1}\) and \(P_r\) are singular, then there is a real nonsingular matrix \(X\) such that \(X^TP_rX=L_{r+1}\).