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

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 [Formula: see text]. In terms of components, such an equivalence means that components R(ijkl) of the first tensor will transform into the components [Formula: see text] of the second tensor for some change of the coordinate system.