Determinant Formulae of Matrices with Certain Symmetry and Its Applications

Abstract

Abstract:In this paper, we introduce formulae for the determinants of matrices with certain symmetry. As applications, we will study the Alexander polynomial and the determinant of a periodic link which is presented as the closure of an oriented 4-tangle.Keywords: determinant of a matrix; Seifert matrix of a link; periodic links; Alexander polynomial of a link MSC: 15A15; 57M25

Computation of Matrix Determinants by Cross-Multiplication: A Rethinking of Dodgson’s Condensation and Reduction by Elementary Row Operations Methods

“…Research literature also offers a plethora of methods, such as Chio's condensation, QR decomposition method, Cholesky decomposition, Dodson's condensation method, Hajrizaj's method, Salihu and Gjonbalaj's method, and Sobamowo's extension of Sarrus' rule to 4 × 4 matrices [2,4,[9][10][11]. Other methods are proposed on the basis of reducing built-in computational errors and specialized matrix forms to include division-free algorithms to compute the determinants of quasi-tridiagonal matrices [12,13], as well as develop determinant formulas for special matrices involving symmetry, such as block matrices [14], break-down free algorithms for computing the determinants of periodic tridiagonal matrices [15], and block diagonalization-based algorithms of block k-tridiagonal matrices [16].…”

Computation of Matrix Determinants by Cross-Multiplication: A Rethinking of Dodgson’s Condensation and Reduction by Elementary Row Operations Methods

“…Research literature also offers a plethora of methods, such as Chio's condensation, QR decomposition method, Cholesky decomposition, Dodson's condensation method, Hajrizaj's method, Salihu and Gjonbalaj's method, and Sobamowo's extension of Sarrus' rule to 4 × 4 matrices [2,4,[9][10][11]. Other methods are proposed on the basis of reducing built-in computational errors and specialized matrix forms to include division-free algorithms to compute the determinants of quasi-tridiagonal matrices [12,13], as well as develop determinant formulas for special matrices involving symmetry, such as block matrices [14], break-down free algorithms for computing the determinants of periodic tridiagonal matrices [15], and block diagonalization-based algorithms of block k-tridiagonal matrices [16].…”