Kronecker Delta

The Kronecker Delta <math>\delta_{ij}</math> satisfies the following property:

<math>\delta_{ij} = \begin{cases}

1 & \mbox{if } i=j \\ 0 & \mbox{if } i \ne j \end{cases}</math>

where <math>i</math> and <math>j</math> are integers. For a summation:

<math>\sum_{i=-\infty}^\infty c_i \delta_{ij} =c_j</math>.

The elements of the identity matrix can be seen as following Kronecker Delta (i.e., <math>(I)_{ij} = \delta_{ij}</math>). To see this, let <math>A</math> be an nxn matrix, <math>a_{ij}</math> be its elements and <math>I</math> be the nxn identity matrix. Then <math>A = AI</math> so,

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

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

The continuous analogue of Kronecker Delta is Dirac delta.

References

Weisstein, Eric W. "Kronecker Delta." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/KroneckerDelta.html