E-Determinants of Tensors
E-Determinants of Tensors
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DOI：
10.1016/j.jsc.2012.10.001
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发表时间：
2011-09
期刊：
影响因子：
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通讯作者：
Shenglong Hu;Zhenghai Huang;C. Ling;L. Qi
中科院分区：
文献类型：
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作者：
Shenglong Hu;Zhenghai Huang;C. Ling;L. Qi
本文研究了张量行列式的性质及其在张量本征值理论中的应用。我们证明了行列式继承了矩阵行列式的许多性质。这些属性包括：多项式系统的可解性，块张量行列式的乘积公式，特征值的乘积公式和Geršgorin不等式。作为一个简单的应用，我们证明了，如果一个多项式系统的首系数张量是一个具有非零对角元素的三角形张量，那么该系统在复空间中一定有解。通过行列式和高阶迹研究了张量的特征多项式。证明了张量的k阶迹等于其特征值的k次幂之和，其特征多项式的系数由高阶迹递归生成.给出了张量二阶迹的显式表达式。
We investigate properties of the determinants of tensors, and their applications in the eigenvalue theory of tensors. We show that the determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, product formula for the determinant of a block tensor, product formula of the eigenvalues and Geršgorinʼs inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution in the complex space. We investigate the characteristic polynomial of a tensor through the determinant and the higher order traces. We show that the k-th order trace of a tensor is equal to the sum of the k-th powers of the eigenvalues of this tensor, and the coefficients of its characteristic polynomial are recursively generated by the higher order traces. Explicit formula for the second order trace of a tensor is given.