Diagonalization of the action matrix and determination of the invariant tensor

Ephra[iuml ]m, M.; Janssen, T.; Janner, A.; Thiers, A.

doi:10.1107/97809553602060000629

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

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International Tables for Crystallography (2006). 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. Janner^c and A. Thiers^d

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.)

References

International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 67-68