A Gentle Introduction to Tensors (2014) [pdf]
I <i>think</i> one of the issues that makes tensors difficult to understand is that different disciplines use the word in different ways. I&#x27;m going to generalize here a bit about how each group uses the term, but I suspect not everyone will agree with me. Apologies in advance!<p>For machine-learning types, a tensor is a multidimensaional array of numbers that admits certain operations (addition, subtraction, hadamard product, etc) [0]<p>For mathematicians, a tensor is a <i>function</i> F. We pass the function an ordered tuple of N vectors and M covectors (or, to keep things simple, N column vectors and M row vectors), and the function returns a scalar. The function F needs to be linear in each of the N vectors and M covectors. In this view, a matrix is a tensor with N=1, M=1. The operations used by machine learning types arise naturally once you crank the mathematical handle a little bit.<p>For physicists, a tensor is what mathematicians would refer to as a <i>tensor field</i>. So they take a space X (think R^N), and for each point in the space they associate a mathematician&#x27;s tensor F as defined above. The properties of the function F are permitted to change from point to point. The Riemannian metric tensor is a classic example of &#x27;tensor&#x27; according to this usage.<p>[0] <a href="https:&#x2F;&#x2F;machinelearningmastery.com&#x2F;introduction-to-tensors-for-machine-learning&#x2F;" rel="nofollow">https:&#x2F;&#x2F;machinelearningmastery.com&#x2F;introduction-to-tensors-f...</a>