If we choose an orthonormal coordinate system ( e 1 , e 2 , e 3 ) we can write the tensor in terms of components with respect to those base vectors as ε ε = ∑ ∑ i = 1 3 ∑ ∑ j = 1 3 ε ε i j e i ⊗ ⊗ e j In matrix form, ε ε _ _ _ _ = [ ε ε 11 ε ε 12 ε ε 13 ε ε 12 ε ε 22 ε ε 23 ε ε 13 ε ε 23 ε ε 33 ] We can easily choose to use another orthonormal coordinate system ( e ^ ^ 1 , e ^ ^ 2 , e ^ ^ 3 ) instead. In that case the components of the tensor are different, say ε ε = ∑ ∑ i = 1 3 ∑ ∑ j = 1 3 ε ε ^ ^ i j e ^ ^ i ⊗ ⊗ e ^ ^ j ⟹ ⟹ ε ε ^ ^ _ _ _ _ = [ ε ε ^ ^ 11 ε ε ^ ^ 12 ε ε ^ ^ 13 ε ε ^ ^ 12 ε ε ^ ^ 22 ε ε ^ ^ 23 ε ε ^ ^ 13 ε ε ^ ^ 23 ε ε ^ ^ 33 ] The components of the strain in the two coordinate systems are related by ε ε ^ ^ i j = ℓ ℓ i p ℓ ℓ j q ε ε p q where the Einstein summation convention for repeated indices has been used and ℓ ℓ i j = e ^ ^ i ⋅ ⋅ e j . In matrix form ε ε ^ ^ _ _ _ _ = L _ _ _ _ ε ε _ _ _ _ L _ _ _ _ T or [ ε ε ^ ^ 11 ε ε ^ ^ 12 ε ε ^ ^ 13 ε ε ^ ^ 21 ε ε ^ ^ 22 ε ε ^ ^ 23 ε ε ^ ^ 31 ε ε ^ ^ 32 ε ε ^ ^ 33 ] = [ ℓ ℓ 11 ℓ ℓ 12 ℓ ℓ 13 ℓ ℓ 21 ℓ ℓ 22 ℓ ℓ 23 ℓ ℓ 31 ℓ ℓ 32 ℓ ℓ 33 ] [ ε ε 11 ε ε 12 ε ε 13 ε ε 21 ε ε 22 ε ε 23 ε ε 31 ε ε 32 ε ε 33 ] [ ℓ ℓ 11 ℓ ℓ 12 ℓ ℓ 13 ℓ ℓ 21 ℓ ℓ 22 ℓ ℓ 23 ℓ ℓ 31 ℓ ℓ 32 ℓ ℓ 33 ] T