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Created page with "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}..."
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|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_of_matrices|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
:<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|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_of_matrices|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