International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 7

## Section 1.1.3.1.2. Tensor product

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France

#### 1.1.3.1.2. Tensor product

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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:

 (i) it is distributive from the right and from the left; (ii) it is associative for multiplication by a scalar; (iii) the tensor products of the vectors with a basis and those with a basis constitute a basis of the product space.

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,