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

TL;DR

This paper investigates the inertia properties of matrices formed by the beta function, revealing their eigenvalue signatures, orthogonality relations, and conditions for invertibility and total positivity.

Contribution

It provides explicit inertia characterizations of beta function matrices and explores their orthogonality, invertibility, and total positivity properties, which are novel insights into their structure.

Findings

Inertia of eta(i,j) is ( /2, 0, n/2) for even n, and ((n+1)/2, 0, (n-1)/2) for odd n.

eta(i,j) is Birkhoff-James orthogonal to the identity matrix I if and only if n is even.

The inverse of eta(i,j) is an integer matrix.

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.