Variance Measures for Symmetric Positive (Semi-) Definite Tensors in Two Dimensions.

Abstract

Calculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor R_abcd. To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor R_abcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors R_abcd and ${\tilde{R}}_{abcd}$. In terms of components, such an equivalence means that components R_ijkl of the first tensor will transform into the components ${\tilde{R}}_{ijkl}$ of the second tensor for some change of the coordinate system.