International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 5
Section 1.1.2.1. Change of basis^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
Let us consider a vector space spanned by the set of n basis vectors , , . The decomposition of a vector using this basis is written using the Einstein convention. The interpretation of the position of the indices is given below. For the present, we shall use the simple rules:
Let us now consider a second basis, . The vector x is independent of the choice of basis and it can be decomposed also in the second basis:
If and are the transformation matrices between the bases and , the following relations hold between the two bases: (summations over j and i, respectively). The matrices and are inverse matrices: (Kronecker symbol: if if ).
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 and 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.