The Lehmer Matrix and its Recursive Analogue

This paper considers the Lehmer matrix and its recursive analogue. The determinant of the Lehmer matrix is derived explicitly by both its LU and Cholesky factorizations. We further define a generalized Lehmer matrix with \((i,j)\) entries \( g_{ij} = \frac{\text{min} \{u_{i+1}, u_{j+1}\}}{\text{max} \{u_{i+1}, u_{j+1}\}} \) where \( u_n \) is the \( n \)th term of a binary sequence \(\{u_n\}\). We derive both the LU and Cholesky factorizations of this analogous matrix and we precisely compute the determinant.