  In linear elasticity, a fourth order elasticity (stiffness) tensor of 21
independent components completely describes deformation properties of a
material. Due to Voigt, this tensor is conventionally represented by a $6\times
6$ symmetric matrix. This classical matrix representation does not conform with
the irreducible decomposition of the elasticity tensor. In this paper, we
construct two alternative matrix representations. The $3\times 7$ matrix
representation is in a correspondence with the permutation transformations of
indices and with the general linear transformation of the basis. An additional
representation of the elasticity tensor by three $3\times 3$ matrices is
suitable for description the irreducible decomposition under the rotation
transformations. We present the elasticity tensor of all crystal systems in
these compact matrix forms and construct the hierarchy diagrams based on this
representation.
