Kronecker Delta

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The Kronecker Delta δij satisfies the following property:

\delta_{ij} = \begin{cases}
1 & \mbox{if } i=j  \\
0 & \mbox{if } i \ne j \end{cases}

where i and j are integers. For a summation:

\sum_{i=-\infty}^\infty c_i \delta_{ij} =c_j.

The elements of the identity matrix can be seen as following Kronecker Delta (i.e, (I)ij = δij). To see this, let A be an nxn matrix, aij be its elements and I be the nxn identity matrix. Then A = AI so,

 a_{ij} = \sum_{k=0}^n a_{ik}(I)_{kj} = \sum_{k=0}^n a_{ik}\delta_{kj} = a_{ij}

where the definition of matrix multiplication and the above property of summation was used.

The continuous analogue of Kronecker Delta is Dirac delta.


Weisstein, Eric W. "Kronecker Delta." From MathWorld--A Wolfram Web Resource.

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