On Generalized Hilbert Matrices

TL;DR

This paper introduces a systematic method for explicitly computing determinants and inverses of generalized Hilbert matrices linked to orthogonal systems, providing explicit formulas and bounds for eigenvalues.

Contribution

It offers a novel systematic approach to compute determinants, inverses, and eigenvalue bounds for generalized Hilbert matrices using explicit orthogonal system representations.

Findings

Explicit formulas for determinants and inverses of generalized Hilbert matrices.

Derived a lower bound for the smallest eigenvalue of these matrices.

Established connections between matrix properties and orthogonal system representations.

Abstract

In this note, we present a systematic method to explicitly compute the determinants and inverses for some generalized Hilbert matrices associated with orthogonal systems with explicit representations. We expressed the determinant, the inverse and a lower bound for the smallest eigenvalue of such matrix in terms of the explicit orthogonal system.