Linear to multi-linear algebra and systems using tensors

Abstract

In the past few decades, multi-linear algebra also known as tensor algebra has been adapted and employed as a tool for various engineering applications. Recent developments in tensor algebra have indicated that several well-known concepts from linear algebra can be extended to a multi-linear setting with the help of a special form of tensor contracted product, known as the Einstein product. Thus, the tensor contracted product and its properties can be harnessed to define the notions of multi-linear system theory where the input, output signals, and the system are inherently multi-domain or multi-modal. This study provides an overview of tensor algebra tools which can be seen as an extension of linear algebra, at the same time highlighting the differences and advantages that the multi-linear setting brings forth. In particular, the notions of tensor inversion, tensor singular value, and tensor eigenvalue decomposition using the Einstein product are explained. In addition, this study also introduces the notion of contracted convolution for both discrete and continuous multi-linear system tensors. Tensor network representation of various tensor operations is also presented. In addition, application of tensor tools in developing transceiver schemes for multi-domain communication systems, with an example of MIMO CDMA system, is presented. This study provides a foundation for professionals whose research involves multi-domain or multi-modal signals and systems.