Spectral behaviour of the matrix \(\big[ f(1- p_i p_j)\big]\) (Q6111063)

Let \(f\) be a concave operator from \((0,\infty)\) to \((0,\infty)\) and let \(0<p_i<1\), \(1\leqslant i\leqslant n\), \(p_i\neq p_j\) for \(i\neq j\). The authors consider the matrix \(A=[f(1-p_ip_j)]_{1\leqslant i,j\leqslant n}\). They prove that it is conditionally negative definite, i.e., it satisfies the condition \(\langle Ax,x\rangle\leqslant 0\) for all \(x\in\left\{x\colon \sum_{i=1}^nx_i=0\right\}\) and: \begin{itemize} \item[1.] If \(f\) is nonlinear, then \(A\) is nonsingular and its inertia (the triple consisting of the numbers of positive, zero and negative eigenvalues) is equal to \((1,0,n-1)\); \item[2.] If \(f\) is linear and \(n>2\), then \(A\) is singular and its inertia is equal to \((1,n-2,1)\). \end{itemize} Moreover, in the case when \(f(t)=\log t\) the matrix \(A\) from the above theorem is negative definite. Additionally, the authors show that the matrix \(A\) may be conditionally negative definite and have inertia \((1,0,n-1)\), although \(f\) is not concave.