decomposable tensor

A tensor – an element in a tensor product of kk vector spaces V 1V 2V kV_1\otimes V_2\otimes\ldots\otimes V_k – is said to be decomposable if it can be written in the form v 1v 1v kv_1\otimes v_1\otimes\ldots\otimes v_k where v iV iv_i \in V_i for i=1,ldots,ki = 1,ldots, k.

If all V iV_i are copies of VV or V *V^* for the same VV then we often talk of vectors in the tensor product as tensors and the tensor product the space of tensors. If V iV_i has a basis {e i s} s=1 n i\{e^s_i\}_{s= 1}^{n_i} then V 1V 2V kV_1\otimes V_2\otimes\ldots\otimes V_k has a basis consisting of all decomposable vectors of the form e 1 s 1e k s ke_1^{s_1}\otimes e_k^{s_k} for all (s 1,,s k)(s_1,\ldots,s_k) such that 1s in i1\leq s_i\leq n_i.

Let VV be finite-dimensional vector space. Then for a tensor AV kA\in V^{\otimes k} we say that AA has decomposability rank rr if

r=min{hA 1,,A hdecomposable,A=A 1++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 AA 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 (