Inertia and other properties of the matrix $\left[β(i,j)\right]$

Abstract

Let $π(A)$, $ξ(A)$ and $ν(A)$, respectively, denote the number of positive, zero and negative eigenvalues of the matrix $A$. Then the triplet $(π(A), ξ(A), ν(A))$ is called the \emph{inertia} of $A$ and is denoted by $\textup{Inertia(A)}$. Let $β$ be the beta function. The inertia of the matrix $\left[β(i,j )\right]$ is shown to be $\left(\frac{n}{2},0,\frac{n}{2}\right)$ if $n$ is even, and $\left(\frac{n+1}{2},0,\frac{n-1}{2}\right)$ if $n$ is odd. %Its connections with Birkhoff-James orthogonality are given. It is also shown that $\left[β(i,j)\right]$ is Birkhoff-James orthogonal to the $n\times n$ identity matrix $I$ in the trace norm if and only if $n$ is even. %We prove that the inverse of $\left[{β(i,j)}\right]$ is an integer matrix. For $0<\la_1<\cdots<\la_n, 0<μ_1<\cdots<μ_n$, it is shown that the matrix $\left[(β(\la_i,μ_j))^m\right]$ is non singular if $μ_{i+1}-μ_{i}\in \N$ for all $1\leq i \leq n-1$. It is also shown that if $μ_{i+1}-μ_i \in \N$ for $1\leq i\leq n-1$, then for $m\in \mathbb N$, the matrix $\left[\frac{1}{β(\la_i,μ_j)^m}\right]$ is totally positive.