International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 7
Section 1.1.3.1.2. Tensor product^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
Let us consider two vector spaces, with n dimensions and with p dimensions, and let there be two linear forms, in and in . We shall associate with these forms a bilinear form called a tensor product which belongs to the product space with np dimensions, :
This correspondence possesses the following properties:
The analytical expression of the tensor product is then One deduces from this that
It is a tensor of rank 2. One can equally well envisage the tensor product of more than two spaces, for example, in npq dimensions. We shall limit ourselves in this study to the case of affine tensors, which are defined in a space constructed from the product of the space with itself or with its conjugate . Thus, a tensor product of rank 3 will have components. The tensor product can be generalized as the product of multilinear forms. One can write, for example,