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

Abstract

Let $\pi(A)$, $\xi(A)$ and $\nu(A)$, respectively, denote the number of positive, zero and negative eigenvalues of the matrix $A$. Then the triplet $(\pi(A), \xi(A), \nu(A))$ is called the \emph{inertia} of $A$ and is denoted by $\textup{Inertia(A)}$. Let $\beta$ be the beta function. The inertia of the matrix $\left[\beta(i,j )\right]$ 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 $\left[\beta(i,j)\right]$ 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 $\left[{\beta(i,j)}\right]$ is an integer matrix. For $0<\la_1<\cdots<\la_n, 0<\mu_1<\cdots<\mu_n$, it is shown that the matrix $\left[(\beta(\la_i,\mu_j))^m\right]$ is non singular if $\mu_{i+1}-\mu_{i}\in \N$ for all $1\leq i \leq n-1$. It is also shown that if $\mu_{i+1}-\mu_i \in \N$ for $1\leq i\leq n-1$, then for $m\in \mathbb N$, the matrix $\left[\frac{1}{\beta(\la_i,\mu_j)^m}\right]$ is totally positive.