Spectral behaviour of the matrix $\left[ f(1- p_ip_j)\right]$

Abstract

Rigorous work on pattern-based special classes of matrices such as $P_r =[(p_i+p_j)^r]$, $B_r= [\mid p_i-p_j\mid ^r]$, etc. shows their spectral behavior and beneficial results in the literature. Bhatia and Jain in 2015 and Dyn, Goodman, and Micchelli in 1986 studied the spectral behavior of $P_r$ and $B_r$ with respect to power function $t\rightarrow t^r$ for distinct positive real numbers $p_1, p_2,\ldots , p_n$ and for positive values of r. Further, Garg and Aujla in 2018 obtained the spectral behavior of the matrices $[f(p_i+p_j)]$ and $[f(\mid p_i-p_j\mid )]$, where f is any operator monotone function from $(0,\infty )$ to $(0,\infty )$. Later in 2021, Tanvi Jain derived the inertia of the matrix $[(1+p_ip_j)^r]$ for distinct positive real values of r. In the present work, we shall show that if f is a nonlinear operator concave function from $(0,\infty )$ to $(0,\infty )$ then the matrix $[f( 1- p_ip_j )]$ is conditionally negative definite, nonsingular, and has inertia $(1,0,n-1)$, however, the result is not the same if f is linear. Further, we will also notice and discuss the change in results in case $f(t)= \log t$. We will give an example to prove the condition, that is, a non-linear function f defined from $(0,\infty )$ to $(0,\infty )$ is operator concave, is not necessary for the matrix $[f(1-p_ip_j)]$, to have the inertia $(1,0,n-1)$. This paper also provides examples of a few functions on the positive real line, which are concave but not operator concave.