PIMS-SFU CSC Distinguished Speaker Seminar: Lek-Heng Lim

This talk is intended for those who, like the speaker, have at some point wondered whether there is a theory of three- or higher-dimensional matrices that parallels matrix theory. A d-dimensional hypermatrix may be viewed as a coordinate representation of an order-d tensor. The simplest type of interactions between two spaces are linear and thus described by 2-tensors (e.g., linear operators, bilinear forms); when there are d > 2 spaces, then the simplest type of interactions are multilinear and these are described by d-tensors. We will see that some of most fundamental notions in computations, mathematics, physics, and statistics are inherently multilinear; higher order tensors and hypermatrices are thus unavoidable in furthering our understanding of these notions. We will discuss how tools like rank, norms, determinant, eigen and singular values can be generalized to hypermatrices, and far from being artificial constructs, they appear naturally in a wide range of phenomena and are enormously useful.