八元数矩阵的行列式及其性质

赋范的可除代数只有四种:实数R,复数C,四元数日和八元数O.由于八元数关于乘法非交换且非结合,如何对八元数矩阵定义行列式并使其具有较好的运算性质变得非常困难.最近,李兴民和黎丽根据"八元数自共轭矩阵的行列式应为实数"这一数学与物理上的需求,通过选择几个八元数乘积的次序和结合方式,首次给出了八元数行列式的定义.但是,与实数、复数以及四元数的相应的情形比较,如此定义的行列式,其所具备的运算性质较少.本文给出了一种新的八元数行列式的定义,它们具备了尽可能多的运算性质,同时使得"八元数自共轭矩阵的行列式为实数"不证自明.

The Determinant for Octonionic Matrix and Its Properties

There are exactly four normed division algebras: the real numbers $R,$ complex numbers $C,$ quaternions $H$ and octonions $O.$ However, due to their noncommtativity and nonassociativity, for a octonionic matrix, how to define the determinant that satisfies nice calculating properties becomes very difficult. Recently, according to the mathematical and physical requirements that ``the determinant of any hermitian octonion matrix should be a real number", by choosing the multiplication orders and the associative methods for $n$ octonioin numbers, the definition of determinant for the octonionic matrix is first given by Li xingmin and Li li. But, compared with the corresponding cases of real, complex and quaternion, such a definition has less calculating properties. In this paper, we give a new definition, which satisfies almost all the properties, and automatically, the determinant for any hermitian octonionic matrix is a real number.