Acta Numerica (2021), pp. 1–208 © Cambridge University Press, 2021 doi:10.1017/S0962492921000076 Printed in the United Kingdom Tensors in computations Lek-Heng Lim Computational and Applied Mathematics Initiative, University of Chicago, Chicago, IL 60637, USA E-mail:
[email protected] The notion of a tensor captures three great ideas: equivariance, multilinearity, separ- ability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way throughthe lensof linear algebra and numerical linear algebra, elucidated with examples from computational and applied mathematics. CONTENTS 1 Introduction 1 2 Tensors via transformation rules 11 3 Tensors via multilinearity 41 4 Tensors via tensor products 85 5 Odds and ends 191 References 195 1. Introduction arXiv:2106.08090v1 [math.NA] 15 Jun 2021 We have two goals in this article: the first is to explain in detail and in the simplest possible terms what a tensor is; the second is to discuss the main ways in which tensors play a role in computations. The two goals are interwoven: what defines a tensor is also what makes it useful in computations, so it is important to gain a genuine understanding of tensors. We will take the reader through the three common definitions of a tensor: as an object that satisfies certain transformation rules, as a multilinear map, and as an element of a tensor product of vector spaces. We will explain the motivations behind these definitions, how one definition leads to the next, and how they all fit together. All three definitions are useful in compu- tational mathematics but in different ways; we will intersperse our discussions of each definition with considerations of how it is employed in computations, using the latter as impetus for the former.