Kronecker Delta

The Kronecker Delta $\delta _{ij}$ satisfies the following property:

$\delta _{ij}={\begin{cases}1&{\mbox{if }}i=j\\0&{\mbox{if }}i\neq j\end{cases}}$

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., $(I)_{ij}=\delta _{ij}$). To see this, let $A$ be an nxn matrix, $a_{ij}$ be its elements and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} be the nxn identity matrix. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = AI} so,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

References

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