Kronecker Delta.

The Kronecker Delta, denoted by $\delta_{ij}$, is defined as

$$\delta_{ij} = \left\{\begin{array}{lll}1 & {\rm if} & i=j,\\ 0 & {\rm if} & i \ne j.\end{array}\right.$$

That is,

$$\begin{split}&\delta_{11} = \delta_{22} = \delta_{33} = 1,\\ ... ..._{21} = \delta_{23} = \delta_{31} = \delta_{32} = 0.\end{split}$$

In other words, the matrix

$$\left[\begin{array}{ccc}\delta_{11} & \delta_{12} & \delta_{13}\\... ...{22} & \delta_{23}\\ \delta_{31} & \delta_{32} & \delta_{33}\end{array}\right]$$

is the identity matrix

$$\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right].$$

We note the following relations

$$\ \ \delta_{ii} = \delta_{11} + \delta_{22} + \delta_{33} = 1 + 1 + 1 = 3,$$

$$\ \ \delta_{1m}a_m = \delta_{11}a_1 + \delta_{12}a_2 + \delta_{13}a_3 = a_1,$$

$$\delta_{2m}a_m = \delta_{21}a_1 + \delta_{22}a_2 + \delta_{23}a_3 = a_2,$$

$$\delta_{3m}a_m = \delta_{31}a_1 + \delta_{32}a_2 + \delta_{33}a_3 = a_3.$$

Or, in general

$$\delta_{im}a_m = a_i.$$

Similarly, one can show that

$$\delta_{im}T_{mj} = T_{ij}.$$

In particular,

$$\delta_{im}\delta_{mj} = \delta_{ij};\ \delta_{im}\delta_{mj}\delta_{jn} = \delta_{in}.$$