# A Note on a Unitary Analog to Redheffer's Matrix

## TL;DR

This paper introduces a unitary analog to Redheffer's matrix, analyzing its determinant and eigenvalues, and expressing its characteristic polynomial coefficients using Stirling numbers, revealing new spectral properties.

## Contribution

It presents a novel unitary matrix analog to Redheffer's matrix, with explicit determinant and eigenvalue characterizations, and connects its polynomial coefficients to Stirling numbers.

## Key findings

- Determinant of the unitary matrix is analogous to Redheffer's matrix.
- The eigenvalue 1 has higher algebraic multiplicity than in the original matrix.
- Characteristic polynomial coefficients relate to Stirling numbers of the second kind.

## Abstract

We study a unitary analog to Redheffer's matrix. It is first proved that the determinant of this matrix is the unitary analogue to that of Redheffer's matrix. We also show that the coefficients of the characteristic polynomial may be expressed as sums of Stirling numbers of the second kind. This implies in particular that $1$ is an eigenvalue with algebraic multiplicity greater than that of Redheffer's matrix.