论文标题
N维空间中M向量的跨产品及其几何意义
The Cross Products of M Vectors in N-dimensional Spaces and Their Geometric Significance
论文作者
论文摘要
在教科书和历史文献中，仅在二维和三维欧几里得空间中定义了跨产品，并且仅在高维欧几里得空间中仅定义了两个向量的跨产品，其度量矩阵是单位矩阵。没有人对高维空间中的任何数量向量的通用定义，其公制矩阵是单位矩阵。实际上，我们还可以在n维空间中定义M向量的交叉产物，在N维空间中，N和M可以将大于1大于1的正整数摄取，并且M不得大于n。在本文中，我们给出了n维空间中M矢量的跨产品的定义，其公式矩阵是任何真实的对称或遗传学矩阵，并提出了两个与矩阵有关的定理，以完美地解释了高维空间中向量的交叉产物的几何含义。此外，交叉产物的长度代表了由M向量跨越的平行多面体的M维体积，并且横乘积的每个组件的绝对值代表体积的每个组件在不同方向上。特别是，在高维的欧几里得和单一空间中，度量矩阵是单位矩阵，该体积的分解仍然满足毕达哥拉利亚定理，并且在n维空间中n个矢量的跨产品是正方形矩阵的决定符，它与这些n vectors形成了这些n vectors as as a aftors as a aftor as afrol as a afl of cow aS cow or as a afrol as a afor a a frow or a frow or a a voR as a frol as a frow or cow or cow or cow or cow or cow or cow ector is a vol.s vectors。我们还解释了度量矩阵及其子膜的决定因素的几何含义，这也可用于理解高维空间中的不变体积元素及其在差分几何形状中的子空间。
In textbooks and historical literature, the cross product has been defined only in 2-dimensional and 3-dimensional Euclidean spaces and the cross product of only two vectors has been defined only in the high dimensional Euclidean space whose metric matrix is the unit matrix. Nobody has given a universal definition for any number of vectors in high dimensional spaces whose metric matrices are the unit matrices. In fact, we can also define the cross product of m vectors in an n-dimensional space, where n and m can take any positive integers larger than 1 and m must not be larger than n. In this paper, we give the definition of the cross product of m vectors in n-dimensional spaces whose metric matrices are any real symmetric or Hermitian matrices, and put forward two theorems related to matrices, so as to perfectly explain the geometric meaning of the cross product of vectors in high dimensional spaces. Furthermore, the length of the cross product represents the m-dimensional volume of the parallel polyhedron spanned by the m vectors, and the absolute value of each component of the cross product represents each component of the volume in different directions. Specially, in the high dimensional Euclidean and unitary space where the metric matrix is the unit matrix, this decomposition of the volume still satisfies the Pythagorean theorem, and the cross product of n vectors in an n-dimensional space is the determinant of the square matrix which is formed with these n vectors as row or column vectors. We also explain the geometric meaning of the determinants of the metric matrices and their sub-matrices, which is also useful for understanding the invariant volume elements in high dimensional spaces and their subspaces in the differential geometry.
