行列式在多重交替线性空间上的高维推广

High Dimensional Extension of Determinant in Multiple Alternating Linear Space

摘要: 从与张量不同的另外角度构造出水平、竖直、侧面三个方向各有n个元素，且沿水平、竖直、侧面三个方向的切片上各有n²个元素，共n³个元素的立方体数阵函数，称为立方行列式，其阶数为n。为方便区分，对只有两个下标的行列式称为狭义行列式。把狭义行列式的全部性质推广至立方行列式以及更高维的超立方行列式。最终给出立方行列式与超立方行列式对应狭义行列式的转化关系和各种变换，并对一般域上的行列式作更深层次的表述。

Abstract: From another angle different from tensor space, we construct a horizontal, vertical and lateral di-rection with n elements, on each slice along the horizontal, vertical and lateral directions have n² elements respectively, cube matrix functions similar to determinant with n³ elements. It is called a cubic determinant and its order is n. For convenience of distinguishing, determinants with only two subscripts are called narrow determinant. Generalize all properties of narrow determinant to cubic determinant and higher dimensional hypercube determinant. In the end, the transformation relation between the cubic determinant and the hypercube determinant corresponding to the narrow determinant is given, and make a more deeper expression of the determinant on the general domain.