Weighted Greatest Common Divisor Matrices:\\ A Complete Characterization

The weighted greatest common divisor (wgcd) has emerged as a fundamental tool in the arithmetic of weighted projective spaces, yet its matrix-theoretic implications remain largely unexplored. In this paper, we introduce and completely characterize Weighted Greatest Common Divisor (WGCD) matrices, whose entries are the wgcds of pairs of tuple elements with their associated weights. We prove the fundamental isomorphism theorem: every WGCD matrix is precisely the classical GCD matrix constructed from its diagonal entries. Specifically, if $\Sigma = (\sigma_{ij})$ is a WGCD matrix with diagonal entries $s_i = \sigma_{ii}$, then $\sigma_{ij} = \gcd(s_i, s_j)$ for all $i,j$. This result generalizes the foundational work of Beslin and Ligh (1989) to the weighted setting and provides a complete reduction of WGCD matrix theory to classical GCD matrix theory. As consequences, we establish that WGCD matrices are positive semidefinite, admit factorization $AA^T$ with entries involving Euler's totient function, and satisfy the determinant inequalities of Li (1990). We also provide efficient algorithms for constructing WGCD matrices using the recent Euclidean-type algorithm for wgcd computation, achieving complexity $O(n^2 \log^3 M)$ where $M = \max_i |x_i|$. This work opens new directions in weighted arithmetic and its applications to projective geometry and cryptography.