Prev Subsection A.1.3: Sum of Tensors Up Section A.1: Second-Order Tensors Subsection A.1.5: Trace of a Second-Order Tensor Next
A.1.4 Dyadic Product of Vectors
The dyadic product of two vectors and is denoted by (or ) and defined as the transformation which satisfies For any , , and , we have thus is a tensor. Its Cartesian components with respect to are In matrix form, Note that in general, , i.e., the dyadic product is not commutative. Also note that thus it is possible to represent a second-order tensor in terms of its Cartesian components in as , or This turns out to be an important result that can be generalized to higher order tensors, e.g., third-order tensors can be represented in terms of their Cartesian components as , and similarly for higher orders.
The scaling transformation derived in a previous example can be represented as