2024/12/18 by Priyanka Grover, Grover, Priyanka, Veer Singh Panwar +1
Computer Science · Mathematics · #Matrix Theory and Algorithms #Mathematics and Applications #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2412.13650
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 inertia of A and is denoted by \textupInertia(A). Let β be the beta function. The inertia of the matrix [β(i,j )] is shown to be ((n)/(2),0,(n)/(2)) if n is even, and ((n+1)/(2),0,(n-1)/(2)) if n is odd. %Its connections with Birkhoff-James orthogonality are given. It is also shown that [β(i,j)] is Birkhoff-James orthogonal to the n× n identity matrix I in the trace norm if and only if n is even. %We prove that the inverse of [β(i,j)] is an integer matrix. For 0<\la1<⋯<\lan, 0<μ1<⋯<μn, it is shown that the matrix [(β(\lai,μj))m] is non singular if μi+1-μi∈ \N for all 1≤ i ≤ n-1. It is also shown that if μi+1-μi ∈ \N for 1≤ i≤ n-1, then for m∈ \mathbb N, the matrix [(1)/(β(\lai,μj)m)] is totally positive.