Hadamard powers of some positive matrices

Abstract

Positivity properties of the Hadamard powers of the matrix $\left[\begin{array}{c}1+x_i{x}_j\end{array}\right]$ for distinct positive real numbers $x_1,\dots ,x_n$ and the matrix $\left[\begin{array}{c}|\cos ((i-j)\pi /n)|\end{array}\right]$ are studied. In particular, it is shown that $\left[\begin{array}{c}(1+x_i{x}_j{)}^r\end{array}\right]$ is not positive semidefinite for any positive real number $r<n-2$ that is not an integer, and $\left[\begin{array}{c}|\cos ((i-j)\pi /n){|}^r\end{array}\right]$ is positive semidefinite for every odd integer $n\ge 3$ and $n-3\le r<n-2.$