Kronecker Delta Symbol

Kronecker delta, denoted by $\delta_{ij}$, is defined as:

\[\delta_{ij}= \begin{cases} 1, & \text{if $ij=j$}.\\ 0, & \text{otherwise}. \end{cases}\]

That is,

$\delta_{11}=1$ $\delta_{12}=0$ $\delta_{13}=0$ $\delta_{21}=0$ $\delta_{22}=1$ $\delta_{23}=0$ $\delta_{31}=0$ $\delta_{32}=0$ $\delta_{33}=1$

Identity Matrix

\[\delta_{ij}\cong \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\]

Properties of the Kronecker Delta

1. Replacement Property:

\begin{align} \delta_{1m}a_m
=\delta_{11}a_1+\delta_{12}a_2+\delta_{13}a_3\
=1a_1+0a_2+0a_3\
=a_1 \end{align}

\begin{align} \delta_{2m}a_m = \delta_{21}a_1+\delta_{22}a_2+0a_3\ \ =a_2 \end{align}

\begin{align} \delta_{3m}a_m & = \delta_{31}a_1+\delta_{32}a_2+\delta_{33}a_3\
=0a_1+0a_2+a_3\
=a_3 \end{align}

So, \(\delta_{im}a_m = a_i\) \((\delta_{1m}a_m , \delta_{2m}a_m , \delta_{3m}a_m) = (a_1, a_2, a_3)\) \(\delta_{1m}am = ai\)

\begin{align} \delta_{1m}T_{mj} & = \delta_{11}T_{1j}+ \delta_{12}T_{2j}+\delta_{13}T_{3j}\ \ =T_{1j} \end{align}

\begin{align} \delta_{2m}T_{mj} & = \delta_{21}T_{1j}+ \delta_{22}T_{2j}+\delta_{23}T_{3j}\ \ =T_{2j} \end{align}

\begin{align} \delta_{3m}T_{mj} & = \delta_{31}T_{1j}+ \delta_{32}T_{2j}+\delta_{33}T_{3j}\ \ =T_{3j} \end{align}
$\begin{bmatrix} \delta_{1m}T_{mj} = T_{ij}\
\delta_{2m}T_{mj} = T_{2j}\
\delta_{3m}T_{mj} = T_{3j}
\end{bmatrix}$ $\Rightarrow$

$\delta_{1m}T_{mj} = T_{1j}, \delta_{2m}T_{mj} = T_{2j}, \delta_{3m}T_{mj}=T_{3j}$

\[\delta_{im}T_{mj} = T_{ij}\]

We can generalize the last two results. If, in a term, the Kronecker delta and another symbol share an index, the index in that symbol can be replaced by the Kronecker delta’s other index in its symbol. For example:

1. $\delta_{im}a_{mb}$ $\Rightarrow$ $\delta_{ib}$
2. $\delta_{mi}a_{mb}$ $\Rightarrow$ $a_{ib}$
3. $\delta_{im}a_{pqms}$ $\Rightarrow$ $a_{pqis}$
4. $\delta_{im}\delta_{mj}$ $\Rightarrow$ $\delta_{ij}$
5. $\delta_{im}\delta_{mn}\delta_{nj}$ $\Rightarrow$ $\delta_{in}\delta_nj$ $\Rightarrow$ $\delta_{ij}$

\begin{align} \delta_{m1}T_{mj} = \delta_{11}T_{1j} + \delta_{21}T_{2j}+ \delta_{31}T_{3j}
= T_{1j}
\end{align}

\begin{align} \delta_{m2}T_{mj} = \delta_{12}T_{1j} + \delta_{22}T_{2j}+\delta_{32}T_{3j}
= T_{2j}
\end{align}

\begin{align} \delta_{m3}T_{mj} = \delta_{13}T_{1j} + \delta_{23}T_{2j}+\delta_{33}T_{3j}
= T_{3j}
\end{align}

\begin{align} \delta_{m1}T_{mj}= T_{1j}
\delta_{m2}T_{mj}= T_{2j}
\delta_{m3}T_{mj}= T_{3j}
\delta_{mi}T_{mj} = T_{ij} \end{align}

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