论文标题
无张量和特征张量的无基础分析
Basis-Free Analysis of Singular Tuples and Eigenpairs of Tensors
论文作者
论文摘要
应用数学中的张量通常定义为数字的多维数组。这假定在$ \ mathbb {r}^n $或其他某些向量空间中的基础选择，并且相应地定义了张力概念。在本文中，我们定义了张量的特征值，特征向量，奇异值和奇异向量，而无需参考基础。无基础方法具有几个优势。首先，它更清楚地显示了张量分析与纯数学领域（例如抽象代数，差异拓扑结构和代数几何形状）之间的关系。其次，它避免了证明根据坐标定义的概念独立于基础的选择。第三，固有的定义通常在概念上更简单。作为插图，我们展示了如何使用来自差异拓扑的莫尔斯理论来分析对称张量的特征值和特征向量。我们还谴责了一些在无基本方法中显而易见的结果，但不是其他方法。
A tensor in applied mathematics is usually defined as a multidimensional array of numbers. This presumes a choice of basis in $\mathbb{R}^n$ or in some other vector space, and tensorial concepts are defined accordingly. In this article we define eigenvalues, eigenvectors, singular values, and singular vectors of a tensor intrinsically, without reference to a basis. The basis-free approach has several advantages. First, it shows more clearly the relationship between tensor analysis and areas of pure mathematics such as abstract algebra, differential topology, and algebraic geometry. Second, it obviates the need to prove that a concept defined in terms of coordinates is independent of the choice of basis. Third, an intrinsic definition is usually conceptually simpler. As illustrations we show how Morse theory from differential topology can be used to analyze eigenvalues and eigenvectors of a symmetric tensor. We also reprove a few results that are obvious in the basis-free approach, but not otherwise.
