Tensor slice ($T(\alpha )=(\alpha \otimes 1\otimes 1)(T)$)

A tensor slice is obtained by contracting one factor of a tensor with a linear functional. For $T\in A\otimes B\otimes C$ and $\alpha \in A^{\ast }$, the contraction $T(\alpha )\in B\otimes C$ is a two-factor tensor and can be displayed as a matrix after choosing bases. Coordinate slices are the contractions with members of a chosen dual basis; their individual matrices depend on that choice.^[ref]

For example, writeThe two first-factor slices are $I_2$ and $N=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$. Contracting with $a{e}^0+b{e}^1$ gives the matrix pencil $a{I}_2+bN$, whose determinant is $a^2$. A change of basis in $A$ replaces the generators of this pencil, but does not change its span.

The contraction map $A^{\ast }\to B\otimes C$ is intrinsic. If $T$ is a sum of $r$ pure tensors, its image is contained in the span of $r$ two-factor pure tensors. Its matrix rank therefore gives a lower bound for tensor rank. Studying slices converts a tensor-decomposition problem into restrictions on a space of matrices, although a single slice rank does not generally determine the tensor rank.