= Tensor slice
{title2=$T(\alpha)=(\alpha\otimes1\otimes1)(T)$}
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 $\alpha\in A^*$, 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.<Geometry and complexity theory by J. M. Landsberg>[section 2.1, tensor notation and tensors regarded as linear maps]{ref}
For example, write
$$
T=e_0\otimes(f_0\otimes g_0+f_1\otimes g_1)+e_1\otimes f_0\otimes g_1.
$$
The two first-factor slices are $I_2$ and $N=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)$. Contracting with $ae^0+be^1$ gives the matrix pencil $aI_2+bN$, 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 $A^*\to B\otimes C$ 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.
Back to article page
