If V → → and W → → are two three dimensional vector operators, then a rank 2 Cartesian dyadic tensors can be formed from nine operators of form T ^ ^ i j = V i ^ ^ W j ^ ^ , U ( R ) † † T ^ ^ i j U ( R ) = U ( R ) † † ( V i ^ ^ W j ^ ^ ) U ( R ) = ( U ( R ) † † V ^ ^ i U ( R ) ) ( U ( R ) † † W ^ ^ j U ( R ) ) = ( ∑ ∑ l = 1 3 R i l V ^ ^ l ⋅ ⋅ ∑ ∑ k = 1 3 R j k W ^ ^ k ) Rearranging terms, we get: U ( R ) † † T ^ ^ i j U ( R ) = ∑ ∑ k = 1 3 ∑ ∑ l = 1 3 ( R i l R j k T ^ ^ l k ) The RHS of the equation is change of basis equation for twice contravariant tensors where the basis are transformed by R − − 1 or the vector components transform by R which matches transformation of vector operator components. Hence the operator tensor described forms a rank 2 tensor, in tensor representation, T ^ ^ = V → → ⊗ ⊗ W → → = ( V ^ ^ i W ^ ^ j ) ( e i ⊗ ⊗ e j ) Similarly, an n-times contravariant tensor operator can be formed similarly by n vector operators.