Recursive Forms for Determinant of K-Tridiagonal Toeplitz Matrices
Toeplitz matrices have garnered renewed interest in recent years due to their practical applications in engineering and computational sciences. Additionally, research has shown their connection to other matrices and their significance in matrix theory. For example, one study demonstrated that any matrix can be expressed as the product of Toeplitz matrices (Ye and Lim, 2016), while another showed that any square matrix is similar to a Toeplitz matrix (Mackey et al., 1999). Numerous studies have examined various properties of Toeplitz matrices, including ideals of lower triangular Toeplitz matrices (Dogan et al., 2018), matrix power computation with band Toeplitz structures (Dogan and Suarez, 2017), and norms of Toeplitz matrices. Moreover, the use of Lucas and Fibonacci numbers has been employed to describe Toeplitz matrix norms (Akbulak and Bozkurt, 2008). With their spectral properties, Toeplitz matrices are crucial to physics, statistics, and signal processing. Furthermore, they aid in the modeling of problems such as computing spline functions, signal and image processing, and polynomial and power series computations (Bini, 1995). This study investigates recursive forms for the determinants of k−tridiagonal Toeplitz matrices. The aim is to extend the known recursions for 1 and 2-tridiagonal Toeplitz matrices. The current research has led to a conjecture on recursive forms for determinants of all k−tridiagonal Toeplitz matrices, k > 2. The study gives a finding of recursions in two forms: one applying Binomial expansion and the other applying LU-decomposition of matrices. The LU-Decomposition is considered, in the Literature, for k−tridiagonal of any matrix but not for Toeplitz matrices. This thesis focused on Toeplitz matrices.
Agyei-Kodie, Eugene, "Recursive Forms for Determinant of K-Tridiagonal Toeplitz Matrices" (2023). ETD Collection for University of Texas, El Paso. AAI30530607.