nLab
decomposable tensor

A tensor – an element in a tensor product of $k$ vector spaces $V_1\otimes V_2\otimes \dots \otimes V_k$ – is said to be decomposable if it can be written in the form $v_1\otimes v_1\otimes \dots \otimes v_k$ where $v_i\in V_i$ for $i=1,\mathrm{ldots},k$.

If all $V_i$ are copies of $V$ or $V^{*}$ for the same $V$ then we often talk of vectors in the tensor product as tensors and the tensor product the space of tensors. If $V_i$ has a basis $\{e_i^s{\}}_{s=1}^{n_i}$ then $V_1\otimes V_2\otimes \dots \otimes V_k$ has a basis consisting of all decomposable vectors of the form $e_1^{s_1}\otimes e_k^{s_k}$ for all $(s_1,\dots ,s_k)$ such that $1\le s_i\le n_i$.

Let $V$ be finite-dimensional vector space. Then for a tensor $A\in V^{\otimes k}$ we say that $A$ has decomposability rank $r$ if

$$r=\min \{h\mid \exists A_1,\dots ,A_h\mathrm{decomposable},A=A_1+\dots +A_h\}$$r = min\{ h | \exists A_1,\ldots, A_h decomposable, A = A_1+\ldots+A_h \}

Distinguish this invariant from the covariance rank of a tensor.

While the decomposability rank of a covariance rank 2 tensor $A$ is the same as the rank of the corresponding matrix, for higher covariance rank tensors we do not have general algorithms how to determine the decomposability rank.