Expression for Rank 2 Tensor in Vector Notation

Question

How does one write the following expression

$D_{jk} (r_k \delta_{ij} - r_{i}\delta_{jk} - r_j \delta_{ik})$

in matrix notation? Is this just

$\textbf{D} (\textbf{r} \times \textbf{I})$?

Answer 1

The given expression in matrix notation should be:

$$ (\textbf{D}-\textbf{D'}- \ \text{trace}(\textbf{D}))\textbf{r} $$

By applying the Kronecker Delta we have: $$ D_{jk} (r_k \delta_{ij} - r_{i}\delta_{jk} - r_j \delta_{ik}) \\ =D_{jk} r_k \delta_{ij} - D_{jk}r_{i}\delta_{jk} - D_{jk}r_j \delta_{ik} \\ =D_{ik} r_k - D_{kk}r_{i} - D_{ji}r_j \\ $$

I was not able to find a cross product (Levi-Civita expression) from the given Kronecker Detla Expression.