Kronecker Delta

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

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

where 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} and $j$ are integers. For a summation:

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 \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.

References

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