In multilinear algebra, a reshaping of tensors is any bijection between the set of indices of an order- tensor and the set of indices of an order- tensor, where. The use of indices presupposes tensors in coordinate representationwith respect to a basis. The coordinate representation of a tensor can be regarded as a multi-dimensional array, and a bijection from one set of indices to another therefore amounts to a rearrangement of the array elements into an array of a different shape. Such a rearrangement constitutes a particular kind of linear map between the vector space of order- tensors and the vectorspace of order- tensors.
Definition
Given a positive integer, the notation refers to the set of the first positive integers. For each integer where for a positive integer, let Vk denote an nk-dimensional vector space over a field. Then there are vector space isomorphisms where is any permutation and is the symmetric group on elements. Via these vector space isomorphisms, a tensor can be interpreted in several ways as an order- tensor where.
The first vector space isomorphism on the list above,, gives the coordinate representation of an abstract tensor. Assume that each of the vector spaces has a basis. The expression of a tensor with respect to this basis has the form where the coefficients are elements of. The coordinate representation of is where is the standard basis vector of. This can be regarded as a d-dimensional array whose elements are the coefficients.
Vectorization
By means of a bijective map, a vector space isomorphism between and is constructed via the mapping where for every natural numbersuch that, the vector denotes the jth standard basis vector of. In such a reshaping, the tensor is simply interpreted as a vector in. This is known as vectorization, and is analogous to vectorization of matrices. A standard choice of bijection is such that which is consistent with the way in which the colon operator in Matlab and GNU Octave reshapes a higher-order tensor into a vector. In general, the vectorization of is the vector.
General flattenings
For any permutation there is a canonical isomorphism between the two tensor products of vector spaces and. Parentheses are usually omitted from such products due to the natural isomorphism between and, but may, of course, be reintroduced to emphasize a particular grouping of factors. In the grouping, there are groups with factors in the group. Letting for each satisfying, an -flattening of a tensor, denoted, is obtained by applying the two processes above within each of the groups of factors. That is, the coordinate representation of the group of factors is obtained using the isomorphism, which requires specifying bases for all of the vector spaces. The result is then vectorized using a bijection to obtain an element of, where, the product of the dimensions of the vector spaces in the group of factors. The result of applying these isomorphisms within each group of factors is an element of, which is a tensor of order. The vectorization of is an -reshaping, wherein.
Matricization
Let be the coordinate representation of an abstract tensor with respect to a basis. A standard factor-k flattening of is an -reshaping in which and. Usually, a standard flattening is denoted by These reshapings are sometimes called matricizations or unfoldings in the literature. A standard choice for the bijections is the one that is consistent with the reshape function in Matlab and GNU Octave, namely
