The Algebra of $S^2$-Upper Triangular Matrices

TL;DR

This paper introduces a new class of matrices called $S^2$-Upper and Lower Triangular Matrices, generalizing traditional triangular matrices, and explores their algebraic properties, including determinant calculation and LU-Decomposition conditions.

Contribution

It defines $S^2$-Triangular Matrices, extends determinant properties, and establishes conditions for LU-Decomposition within this new matrix class.

Findings

Determinant of $S^2$-Upper Triangular Matrices equals the product of diagonal entries.

Constructs the algebra of $S^2$-Upper Triangular Matrices.

Provides conditions for LU-Decomposition using $S^2$-Lower and $S^2$-Upper Triangular Matrices.

Abstract

Based on work presented in [4], we define $S^2$-Upper Triangular Matrices and $S^2$-Lower Triangular Matrices, two special types of $d\times d(2d-1)$ matrices generalizing Upper and Lower Triangular Matrices, respectively. Then, we show that the property that the determinant of an Upper Triangular Matrix is the product of its diagonal entries is generalized under our construction. Further, we construct the algebra of $S^2$-Upper Triangular Matrices and give conditions for an LU-Decomposition with $S^2$-Lower Triangular and $S^2$-Upper Triangular Matrices, respectively.