Handbook of Linear Algebra, 2nd Edition

Chapter 15

Tensors and Hypermatrices

Lek-Heng Lim

University of Chicago

Most chapters in this handbook are concerned with various aspects and implications of linearity; Chapter 14 and this chapter are unusual in that they are about multilinearity. Just as linear operators and their coordinate representations, i.e., matrices, are the main objects of interest in other chapters, tensors and their coordinate representations, i.e., hypermatrices, are the main objects of interest in this chapter. The parallel is summarized in the following schematic:

$linearity\to linearoperators,bilinearforms,dyads\to matricesmultilinearity\to tensors\to hypermatrices$

Chapter 14, or indeed the monographs on multilinear algebra such as [Gre78, Mar23, Nor84, Yok92], are about ...