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Section 8.1 Kronecker Delta
Definition 8.1.1 . The Kronecker Delta.
The Kronecker Delta is a simple switch that compares two values \(i\) and \(j\) in some discrete set. The switch turns on when \(i\) is the same as \(j\) and off when they are different, i.e.,
\begin{equation}
\delta_{ij}
= \begin{cases}0\quad \amp i\not= j\\
1\quad\amp i=j
\end{cases}\tag{8.1.1}
\end{equation}
Example 1: Kronecker Delta in a Sum.
The most common place that the Kronecker delta appears is inside a sum. In this case, it can be used to collapse the sum to a single term. For example:
\begin{align}
\sum_{i=1}^{\infty} a_i\, \delta_{i3}
\amp = a_1 \cancelto{0}{\delta_{13}}
+ a_2 \cancelto{0}{\delta_{23}} +
a_3 \cancelto{1}{\delta_{33}} +\dots\tag{8.1.2}\\
\amp =a_3\tag{8.1.3}
\end{align}
In this example the factor of \(\delta_{i3}\) is zero in every term except the one when \(i=3\text{,}\) so all of those terms are zero and do not contribute to the sum. When \(i=3\text{,}\) the factor of \(\delta_{i3}\) is just one, so that is the only term that contributes.
Example 2: Trace of a Matrix with Kronecker Deltas.
If you want to add up all of the elements \(a_{ij}\) of a matrix \(A\) (where, by convention, the first index \(i\) labels the rows and the second index \(j\) labels the columns), this requires a double sum, one for each index.
\begin{equation}
\sum_{i=1}^{3} \sum_{i=j}^{3}a_{ij}\tag{8.1.4}
\end{equation}
If, instead, you want to find the trace of the matrix \(\mathrm{tr} A\text{,}\) then you only want to add up the diagonal elements. A Kronecker delta in the double sum will ensure that you only take the elements for which the row and column labels are the same.
\begin{align}
\mathrm{tr} A\amp\doteq
\sum_{i=1}^{3} \sum_{i=j}^{3}
a_{ij}\delta_{ij}\tag{8.1.5}\\
\amp = \sum_{i=1}^{3} a_{ii}\tag{8.1.6}
\end{align}
Prerequisites 8.1.2 . Try it yourself: The Identity Matrix.
Write the elements of the \(3\times 3\) identity matrix in terms of the Kronecker delta.
Answer .
\begin{equation}
I\doteq \begin{pmatrix}
\delta_{11}\amp \delta_{12}\amp \delta_{13}\\
\delta_{21}\amp \delta_{22}\amp \delta_{23}\\
\delta_{31}\amp \delta_{32}\amp \delta_{33}
\end{pmatrix}\tag{8.1.7}
\end{equation}
