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0, & \mbox{if } i \ne j \end{matrix}\right.</math>
==თვისებები==
==Properties of the delta function==
<!-- Please do not "correct" sifting to shifting. The Kronecker delta acts as a sieve; that is, it *sifts*. -->
კრონეკერის სიმბოლოს აქვს ე.წ. წანაცვლების თვისება, ანუ ნებისმიერი <math>j\in\mathbb Z</math>-სთვის:
The Kronecker delta has the so-called ''sifting'' property that for <math>j\in\mathbb Z</math>:
:<math>\sum_{i=-\infty}^\infty a_i \delta_{ij} =a_j.</math>
and if the integers are viewed as a [[measure space]], endowed with the [[counting measure]], then this property coincides with the defining property of the [[Dirac delta function]]
:<math>\int_{-\infty}^\infty \delta(x-y)f(x) dx=f(y),</math> ▼and in fact Dirac's delta was named after the Kronecker delta because of this analogous property. In signal processing it is usually the context (discrete or continuous time) that distinguishes the Kronecker and Dirac "functions". And by convention, <math>\delta(t)\,</math> generally indicates continuous time (Dirac), whereas arguments like ''i'', ''j'', ''k'', ''l'', ''m'', and ''n'' are usually reserved for discrete time (Kronecker). Another common practice is to represent discrete sequences with square brackets; thus: <math>\delta[n]\,</math>. It is important to note that the Kronecker delta is not the result of directly sampling the Dirac delta function.
The Kronecker delta is used in many areas of mathematics.
===Linear algebra===
In [[linear algebra]], the [[identity matrix]] can be written as <math>(\delta_{ij})_{i,j=1}^n\,</math>.
If it is considered as a [[tensor]], the Kronecker tensor, it can be written
<math>\delta^i_j</math> with a [[covariance and contravariance of vectors|covariant]] index ''j'' and [[Covariance and contravariance of vectors|contravariant]] index ''i''.
ეს თვისება ანალოგიურია [[დირაკის დელტა ფუნქცია|დირაკის დელტა ფუნქციის]] შემდეგი თვისებისა.
This (1,1) tensor represents:
▲:<math>\int_{-\infty}^\infty \delta(x-y)f(x) dx=f(y) ,.</math>
* The identity matrix, considered as a [[linear mapping]]
* The [[trace (linear algebra)|trace]]
* The [[inner product]] <math>V^* \otimes V \to K</math>
* The map <math>K \to V^* \otimes V</math>, representing scalar multiplication as a sum of [[outer product]]s.
==Relationship to the [[Dirac delta function]]==
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