Kronecker Delta

Relevant parts to questions…

• Using ${\delta }_{ij}{a}_j=a_i$.
• Using ${\delta }_{ii}=3$, if in three-dimensions (convention).

The Kronecker delta, or substitution tensor, is defined by ${\delta }_{ij}=\{\begin{array}{ll}1& \text{if }i=j\\ 0& \text{if }i\ne j\end{array}$, where $i$ and $j$ can be 1, 2, or 3. This is effectively equivalent to the identity matrix:

$${\delta }_{ij}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

The Kronecker delta hence has properties like its continuous equivalent, the Dirac delta-function::It replaces the repeated index with the free index, ${\delta }_{ij}{a}_j=a_i$, or symmetrically (using relabelling/reordering), ${\delta }_{ji}{a}_i=a_j$.

This hence allows us to define the dot product as $\mathbf{a}\cdot \mathbf{b}=a_i{b}_i=a_i({\delta }_{ij}{b}_j)={\delta }_{ij}{a}_i{b}_j$, and vice versa.