Tensor Analysis: Spectral Theory and Special Tensors
Tensor Analysis: Spectral Theory and Special Tensors
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DOI：
10.1137/1.9781611974751
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发表时间：
2017-04
期刊：
影响因子：
--
通讯作者：
L. Qi;Ziyan Luo
中科院分区：
文献类型：
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作者：
L. Qi;Ziyan Luo
矩阵理论是数学和科学中最基本的工具之一，已经有许多关于矩阵分析的经典著作来探索这一理论。作为矩阵的高阶推广，张量或超矩阵的概念由于多指标数据集在科学和工程领域的广泛应用而被引入和研究。与矩阵相比，张量有更多的下标，具有它们自己的几何和代数结构，如果我们将它们重塑或展开成矩阵，可能会丢失这些结构。它们严重依赖于张量结构的固有特征之一是张量本征值的概念，它比矩阵情况下的张量本征值要复杂得多。因此，张量必须被视为数据对象本身，并且需要关于这种新类型对象的理论，而矩阵分析仍然很重要，但不那么重要。
Matrix theory is one of the most fundamental tools of mathematics and science, and a number of classical books on matrix analysis have been written to explore this theory. As a higher order generalization of a matrix, the concept of tensors or hypermatrices has been introduced and studied due to multi-indexed data sets from wide applications in scientific and engineering communities. With more subscripts, compared to matrices, tensors possess their own geometric and algebraic structures which might be lost if we reshape or unfold them into matrices. One of their intrinsic features that heavily relies on the tensor structures is the concept of tensor eigenvalues, which turns out to be much more complex than that of the matrix case. Thus, tensors must then be treated as data objects in their own right, and theory on this new type of objects is required, while matrix analysis is still of importance but less so.