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

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

Section Behaviour under a change of basis

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Behaviour under a change of basis

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A multilinear form is, by definition, invariant under a change of basis. Let us consider, for example, the trilinear form ([link]. If we change the system of coordinates, the components of vectors x, y, z become [x^{i} = B_{\alpha}^{i} x'^{\alpha };\quad y_{j}= A_{j}^{\beta} y'_{\beta};\quad z^{k}= B_{\gamma}^{k}z'^{\gamma}.]

Let us put these expressions into the trilinear form ([link]: [P({\bf x}, {\bf y}, {\bf z}) = p\hskip1pt_{ik}^{j} B_{\alpha }^{i} x'^{\alpha}A_{j}^{\beta} y'_{\beta}B_{\gamma}^{k} z'^{\gamma}.]

Now we can equally well make the components of the tensor appear in the new basis: [P({\bf x}, {\bf y}, {\bf z}) = p'^{\beta}_{\alpha \gamma} x'^{\alpha}y'_{\beta }z'^{\gamma}.]

As the decomposition is unique, one obtains [p'^{\beta}_{\alpha \gamma} = p\hskip1pt_{ik}^{j} B_{\alpha}^{i} A_{j}^{\beta} B^{k}_{\gamma}. \eqno{(}]

One thus deduces the rule for transforming the components of a tensor q times covariant and r times contravariant: they transform like the product of q covariant components and r contravariant components.

This transformation rule can be taken inversely as the definition of the components of a tensor of rank [n = q + r].

Example.  The operator O representing a symmetry operation has the character of a tensor. In fact, under a change of basis, O transforms into O′: [O' = A O A^{-1}]so that [O'^{i}_{j} = A_{k}^{i} O_{l}^{k} (A^{-1})_{j}^{l}.]Now the matrices A and B are inverses of one another: [O'^{i}_{j} = A_{k}^{i} O_{l}^{k} B_{j}^{l}.]The symmetry operator is a tensor of rank 2, once covariant and once contravariant.

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