Tensors.jl
A tensor $T$ of order[1] $n$ is a multilinear[2] application between $n$ vector spaces over a field $\mathcal{F}$.
\[T : \mathcal{F}^{\dim(1)} \times \dots \times \mathcal{F}^{\dim(n)} \mapsto \mathcal{F}\]
In layman's terms, it is a linear function whose inputs are vectors and the output is a scalar number.
\[T(\mathbf{v}^{(1)}, \dots, \mathbf{v}^{(n)}) = c \in \mathcal{F} \qquad\qquad \forall i, \mathbf{v}^{(i)} \in \mathcal{F}^{\dim(i)}\]
Tensor algebra is a higher-dimensional generalization of linear algebra, where scalar numbers can be viewed as order-0 tensors, vectors as order-1 tensors, matrices as order-2 tensors, ...
Letters are used to identify each of the vector spaces the tensor relates to. In computer science, you would intuitively think of tensors as "n-dimensional arrays with named dimensions".
\[T_{ijk}\]
- 1The order of a tensor may also be known as rank or dimensionality in other fields. However, these can be missleading, since it has nothing to do with the rank of linear algebra nor with the dimensionality of a vector space. We prefer to use order.
- 2Meaning that the relationships between the output and the inputs, and the inputs between them, are linear.