n维空间正交张量的典则表示和自由度公式

Canonical Representations and Degree of Freedom Formulae of Orthogonal Tensors in n-Dimensional Euclidean Space

本文借助于正交张量特征值的特性,采用剖分的方法.利用二维正交张量典则表示,很快就构造出一般n维欧氏空间上的正交张量的典则表示.利用Cayley-Hamilton定理,求得了正交张量各主不变量之间的相关方程,从而使得正交张量特征根的求解只需要在一个阶数不大于空间维数n的一半的代数方程上进行.本文还给出了正交张量的独立参数个数——自由度的计算公式.

In this paper, with the help of the eigenvalue properties of orthogonal tensors in n-dimensional Euclidean space and the representations of the orthogonal tensors in 2-dimensional space, the canonical representations of orthogonal tensors in n-dimensional Euclidean space are easily gotten. The paper also gives all the constraint relationships among the principal invariants of arbitrarily given orthogonal tensor by use of Cayley-Hamilton theorem; these results make it possible to solve all the eigenvalues of any orthogonal tensor based on a quite reduced equation of m-th order, where m is the integer part of n/2. Finally, the formulae of the degree of freedom of orthogonal tensors are given.