Akademik Veri Yönetim Sistemi
TROIA 3RD INTERNATIONAL CONFERENCE ON APPLIED SCIENCES, Çanakkale, Türkiye, 27 - 28 Şubat 2026, ss.12-29, (Tam Metin Bildiri)
Bidiagonal matrices represent one of the simplest yet most structurally rich classes of matrices in numerical linear algebra. Despite having only \(2n - 1\) independent parameters, their algebraic, spectral, and perturbation properties make them fundamental in both theoretical investigations and practical computations. This study investigates the deep connections between bidiagonal matrices and Hyper \(G\)-matrix structures, highlighting how inverse–transpose duality under diagonal scaling forms canonical Hyper \(G\)-pairs. We rigorously establish several new results, including componentwise perturbation bounds, checkerboard sign patterns for inverses with positive entries, and preservation of monotone eigenvalue order under products of positive bidiagonal matrices. The implications of these results extend to numerical stability, condition number estimation, and error analysis for linear systems. Explicit constructions of scaling matrices and product-based formulations allow for efficient computation of inverses, condition numbers, and componentwise error bounds in \(O(n)\) or \(O(kn)\) operations, avoiding the explicit formation of large matrices. Furthermore, we demonstrate applications of these findings in the context of totally positive matrices, Vandermonde and Pascal-type matrices, singular value decomposition, and iterative algorithms such as LSQR and Lanczos methods. Overall, this work bridges classical total positivity theory with modern matrix symmetry and Hyper \(G\)-matrix frameworks, providing a unified perspective that informs both theoretical research and algorithmic design.