Matrix-Theoretic Investigation of Generalized Geometric Frank Matrices: Spread Estimates and Arithmetic Insights

This study introduces a novel generalization: the generalized geometric Frank matrix, which extends the classical Frank matrix and its known variants. We systematically examine its algebraic structure, providing detailed analyses of its factorizations, determinant, inverse, and various norm computations. Furthermore, we investigate the reciprocal form of the reciprocal generalized geometric Frank matrix and reveal a variety of its intriguing algebraic properties. To illustrate the applicability of our theoretical results, we present a compelling example using Fibonacci number entries within the Frank matrix framework. Additionally, we analyze how the spread’s upper bounds are influenced by variations in the parameter r and the matrix dimension. Also, to formally assess the computational implications of these structural choices, we use Big O notation to describe how the computational cost scales with the matrix size n and the iteration count k(r). Our findings demonstrate that selecting r < 1 and utilizing lower-dimensional generalized geometric Frank matrices can yield tighter bounds and significantly reduce computational complexity. These results highlight the potential of the proposed matrix class for optimization problems where efficiency is critical.