Coordinate-free Tensor Analysis
Vensi Basso, Joao Marcos
2020
- In applied mathematics, tensors are usually viewed as multidimensional arrays of numbers, generalizing the 2-dimensional matrices. They are important in many areas of mathematics; for example, in data analysis if a function depends on three parameters, then a set of data for the function would be described by a 3-dimensional array of numbers. The interpretation of tensors as arrays implicitly ... read moreassumes a choice of basis. In this work we step back to the abstract definition of tensors and seek coordinate-independent generalizations of matrix concepts such as eigenvalues, eigenvectors, and rank. These explorations lead to tensor eigenvalues, tensor eigenvectors, tensor rank and border rank as well as connections to algebra, manifolds, algebraic geometry, and complexity theory. The advantages of the coordinate-free approach are that it is conceptually simpler, reveals insights, and allows easier generalizations. Our coordinate-free definitions of the eigenvalues and eigenvectors of a symmetric tensor appear to be new. Advisor: Professor Loring W. Tu.read less
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