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

International Tables for Crystallography (2013). Vol. D, ch. 1.2, pp. 67-68

Section 1.2.7.4.3. Diagonalization of the action matrix and determination of the invariant tensor

M. Ephraïm,b T. Janssen,a A. Jannerc and A. Thiersd

1.2.7.4.3. Diagonalization of the action matrix and determination of the invariant tensor

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An invariant element of the tensor space under the group G is a vector v that is left invariant under each generator: [\pmatrix{ M_{1}-E \cr M_{2}-E \cr \vdots \cr M_{s}-E } v = \Omega v = 0. ]If the number of generators is one, [\Omega =M-E]. This equation is solved by diagonalization: [P\Omega Q Q^{-1} v = D Q^{-1}v = 0, ]where [D_{ij}=d_{i}\delta_{ij}]. The dimension of the solution space is the number of elements [d_{i}] that are equal to zero. The corresponding rows of Q form a basis for the solution space. (See example further on.)








































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