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 $\text{Inertia(A)}$. 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 $0<\la_1<\cdots<\la_n, 0<μ_1<\cdots<μ_n$, it is shown that the matrix $\left[(β(\la_i,μ_j))^m\right]$ is non singular if ${\mu }_{i+1}-{\mu }_i\in \mathbb{N}$ for all $1\le i\le n-1$. It is also shown that if ${\mu }_{i+1}-{\mu }_i\in \mathbb{N}$ for $1\le i\le n-1$, then for $m\in \mathbb{N}$, the matrix $\left[\frac{1}{β(\la_i,μ_j)^m}\right]$ is totally positive.

This manuscript contains 11 pages