In mathematicsthe Kronecker delta named after Leopold Kronecker is a function of two variablesusually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:.
The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above. The restriction to positive integers is common, but there is no reason it cannot have negative integers as well as positive, or any discrete rational numbers.
If i and j above take rational values, then for example. This latter case is for convenience. However, the Kronecker delta is not defined for complex numbers. This can be derived using the formula for the finite geometric series. Sometimes the Kronecker delta is called the substitution tensor.
The function is referred to as an impulseor unit impulse. When it is the input to a discrete-time signal processing element, the output is called the impulse response of the element.
In signal processing it is usually the context discrete or continuous time that distinguishes the Kronecker and Dirac "functions". The Kronecker delta is not the result of directly sampling the Dirac delta function.
The Kronecker delta forms the multiplicative identity element of an incidence algebra. In probability theory and statisticsthe Kronecker delta and Dirac delta function can both be used to represent a discrete distribution.
Equivalently, the probability density function f x of the distribution can be written using the Dirac delta function as. Under certain conditions, the Kronecker delta can arise from sampling a Dirac delta function.
For example, if a Dirac delta impulse occurs exactly at a sampling point and is ideally lowpass-filtered with cutoff at the critical frequency per the Nyquist—Shannon sampling theoremthe resulting discrete-time signal will be a Kronecker delta function.
The generalized Kronecker delta or multi-index Kronecker delta of order 2 p is a type pp tensor that is a completely antisymmetric in its p upper indices, and also in its p lower indices.
Two definitions that differ by a factor of p!The tensor functions discrete delta and Kronecker delta first appeared in the works L. Kroneckerand T.
Levi—Civita For all possible values of their arguments, the discrete delta functions andKronecker delta functions andand signature Levi—Civita symbol are defined by the formulas:. In other words, the Kronecker delta function is equal to 1 if all its arguments are equal.
In the case of one variable, the discrete delta function coincides with the Kronecker delta function. In the case of several variables, the discrete delta function coincides with Kronecker delta function :. Connections within the group of tensor functions and with other function groups. The tensor functions,and have the following representations through equivalent functions:.
The best-known properties and formulas of the tensor functions. The tensor functions,and can have unit values at infinity:.Index/Tensor Notation: Reducing The Kronecker Delta - Lesson 3
The tensor functions, and have the following values for some specialized variables:. Their possible values are and. The tensor functions, and do not have periodicity.
The tensor functions,and are even functions:. The tensor functions, and have permutation symmetry, for example:. The discrete delta function and Kronecker delta function have the following integral representations along the interval and unit circle :. The tensor functions, and satisfy various identities, for example:. The tensor functions, and have the following complex characteristics:.
Differentiation of the tensor functions and can be provided by the following formulas:. Indefinite integration of the tensor functions and can be provided by the following formulas:. The following relations represent the sifting properties of the Kronecker and discrete delta functions:.
It only takes a minute to sign up. I am trying to introduce a rule of multiplication of entries of a matrix which would have the following property:. How can I introduce such rule in Mathematica? Of course, the multiplication of entries which have not equal indexes should be usual one. Note expand is needed so the products appear explicitly.
This is going to break if you have certain more complicated expressions such as products with more than two deltas. I expect this can be handled using Simplify with TransformationFunctions as well. That would obviate the need for Expand and leave expressions alone if no simplification is possible.
Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Rule for multiplying of Kronecker-delta-type functions Ask Question. Asked 4 years, 1 month ago. Active 4 years, 1 month ago. Viewed times. UPD Also delta is symmetric. Dmitri Dmitri 2 2 silver badges 9 9 bronze badges.
If so, can you provide more details about your actual use-case? It's a little unclear exactly what you would need. Something like a replacement Rule that looks inside expressions and sees when arguments to delta s overlap could work, but we need more info. For instance, are you manipulating symbolic matrices defined with these delta s? Are repeated indices implicitly summed over? It seems that you have now asked 8 questions, many of which have excellent answers, but you have yet to accept any of them.
While it is the intent of this site and it is the motivation of the majority of its users to provide answers to general questions that many people not just the asker can benefit from, it is still the case that answerers like to be thanked by having an answer that they spent time on accepted. Please consider spending a few seconds to either accept answers that answered your questions or commenting on those answers. Active Oldest Votes. It seems that it works even for some products with more factors.
Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.The simplest interpretation of the Kronecker delta is as the discrete version of the delta function defined by. The Kronecker delta is implemented in the Wolfram Language as KroneckerDelta [ ij ], as well as in a generalized form KroneckerDelta [ ijTechnically, the Kronecker delta is a tensor defined by the relationship. Since, by definition, the coordinates and are independent for.
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Well, the kronecker delta returns, given two indices, either 1 or 0, so just a number. An identity matrix of size nxn is of course a matrix, not just a number! But the Kronecker delta considered as a whole is no more a single number than a matrix of numbers is a single number!
Tensor Notation (Basics)
The elements of an identity matrix are equivalent to the elements of the Kronecker delta, so why make the difference? Homework Helper. The Kronecker delta does not have elements. It is not a matrix. It is a function it takes as input the pair i,j and returns 1 if they are the same and zero otherwise. The identity matrix is a matrix, the Kronecker delta is not. There is no simpler way to say it than that. So the only difference is in terminology?
They even have the same notation! No, one is a linear map the other is not a linear map, that is not just a terminological difference: they are completely different things. They are just different things. You can use one of them to describe the coordinate functions on the other but that does not make them equal in any sense. The kronecker delta is what you use when you want to work componentwise with matrices. You can make a matrix from it in the obvious way and that will be the identity. That does not make it the identity matrix.
It is not the identity matrix. Last edited: May 25, No, not only terminilogy. What I was trying to point out - and matt as well I believe - is that there is a fundamental difference.
The identity matrix is a matrix and consists of elements, as matt saidthe kronecker delta is not. You must log in or register to reply here.Documentation Help Center. For kroneckerDelta with numeric inputs, use the eq function instead. Set symbolic variable m equal to symbolic variable n and test their equality using kroneckerDelta.
Note that kroneckerDelta p, q is equal to kroneckerDelta p - q, 0.
To force a logical result for undecidable inputs, use isAlways. The isAlways function issues a warning and returns logical 0 false for undecidable inputs.
Set the Unknown option to false to suppress the warning.
Set symbolic variable m to 0 and test m for equality with 0. The kroneckerDelta function errors because it does not accept numeric inputs of type double. Use sym to convert 0 to a symbolic object before assigning it to m. This is because kroneckerDelta only accepts symbolic inputs. Note that kroneckerDelta m is equal to kroneckerDelta m, 0.
Compare a vector of numbers [1 2 3 4] with symbolic variable m. Set m to 3. The third element of sol is 1 indicating that the third element of V equals m. Compare matrices A and B. Compare A and B using kroneckerDelta. The elements of sol that are 1 indicate that the corresponding elements of A and B are equal.
The elements of sol that are 0 indicate that the corresponding elements of A and B are not equal. Use this output as input to ztrans to return the initial input expression.
Use filter to find the response of a filter when the input is the Kronecker Delta function. Convert k to a symbolic vector using sym because kroneckerDelta only accepts symbolic inputs, and convert it back to double using double. Provide arbitrary filter coefficients a and b for simplicity. Input, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, or multidimensional array. At least one of the inputs, m or nmust be symbolic.
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The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Kronecker delta matrix question Ask Question. Asked 6 years, 5 months ago. Active 6 years, 5 months ago. Viewed 1k times. Ryan Ryan 21 2 2 bronze badges. Active Oldest Votes. Give it a try.
Can you explain this to me? See if it helps. Much appreciated, I understand it now. Shaun Shaun Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.
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