Theory and Computation of Tensors

First, the tensors here are not the tensors of physics; they are the tensors of Big Data, and are just big multidimensional arrays of numbers. We know that matrices have eigenvalues and eigenvectors that reveal important properties of the matrices. A lot of matrix features, such as eigenvectors, singular value decomposition (SVD), spectral radius, and the Gershgorin circle theorem can be extended to tensors. The present book is primarily about these extensions and numerical methods for how to estimate and calculate them.

Roughly half the book is devoted to defining and developing properties of tensors, and the other half to algorithms. The book is weak on applications and examples, and there are no exercises, although it does call out application areas where the techniques are useful. Most of the algorithmic part deals with tensors of two particular forms, the Hankel tensors and the \(\mathcal{M}\)-tensors.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.