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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 5

Section Change of basis

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: Change of basis

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Let us consider a vector space spanned by the set of n basis vectors [{\bf e}_{1}], [{\bf e}_{2}], [{\bf e}_{3},\ldots, {\bf e}_{n}]. The decomposition of a vector using this basis is written [{\bf x} = x^{i}{\bf e}_{i} \eqno(]using the Einstein convention. The interpretation of the position of the indices is given below. For the present, we shall use the simple rules:

  • (i) the index is a subscript when attached to basis vectors;

  • (ii) the index is a superscript when attached to the components. The components are numerical coordinates and are therefore dimensionless numbers.

Let us now consider a second basis, [{\bf e}'_{j}]. The vector x is independent of the choice of basis and it can be decomposed also in the second basis: [{\bf x} = x'^{i}{\bf e}'_{i}. \eqno(]

If [A\hskip1pt_{i}^{j}] and [B_{j}^{i}] are the transformation matrices between the bases [{\bf e}_{i}] and [{\bf e}'_{j}], the following relations hold between the two bases: [\left.\matrix{{\bf e}_{i} = A\hskip1pt_{i}^{j}{\bf e}'_{j}\semi\hfill &{\bf e}'_{j} = B_{j}^{i}{\bf e}_{i}\hfill \cr x^{i} = B_{j}^{i} x'^{j}\semi &x'^{j} = A\hskip1pt_{i}^{j} x^{i}\cr}\right\} \eqno(](summations over j and i, respectively). The matrices [A\hskip1pt_{i}^{j}] and [B_{j}^{i}] are inverse matrices: [A\hskip1pt_{i}^{j} B_{j}^{k} = \delta_{i}^{k} \eqno(](Kronecker symbol: [\delta_{i}^{k} = 0] if [i \neq k, = 1] if [i = k]).

Important Remark. The behaviour of the basis vectors and of the components of the vectors in a transformation are different. The roles of the matrices [A\hskip1pt_{i}^{j}] and [B_{j}^{i}] are opposite in each case. The components are said to be contravariant. Everything that transforms like a basis vector is covariant and is characterized by an inferior index. Everything that transforms like a component is contravariant and is characterized by a superior index. The property describing the way a mathematical body transforms under a change of basis is called variance.

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