Tensor (intrinsic Definition) - Tensor Rank

Tensor Rank

The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix. A matrix thus has rank one if it can be written as an outer product:

More generally, the rank of a matrix is the length of the smallest decomposition of a matrix A into a sum of such outer products:

Similarly, a tensor of rank one (also called a simple tensor) is a tensor that can be written as a tensor product of the form

where a, b,...,d are in V or V*. That is, if the tensor is completely factorizable. In indices, a tensor of rank 1 is a tensor of the form

Every tensor can be expressed as a sum of rank 1 tensors. The rank of a general tensor T is defined to be the minimum number of rank 1 tensors with which it is possible to express T as a sum (Bourbaki 1988, II, §7, no. 8).

A nonzero order 1 tensor always has rank 1. The zero tensor has rank zero. The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix (Halmos 1974, §51), and can be determined from Gaussian elimination for instance. The rank of an order 3 or higher tensor is however often very hard to determine, and low rank decompositions of tensors are sometimes of great practical interest (de Groote 1987). Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of polynomials can be recast as the problem of simultaneously evaluating a set of bilinear forms

for given inputs x_i and y_j. If a low-rank decomposition of the tensor T is known, then an efficient evaluation strategy is known (Knuth 1998, pp. 506–508).