If all are copies of or for the same then we often talk of vectors in the tensor product as tensors and the tensor product the space of tensors. If has a basis then has a basis consisting of all decomposable vectors of the form for all such that .
Let be finite-dimensional vector space. Then for a tensor we say that has decomposability rank if
Distinguish this invariant from the covariance rank of a tensor.
While the decomposability rank of a covariance rank 2 tensor 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.
Revised on September 2, 2011 00:02:55
by Urs Schreiber