Difference between Kronecker delta and identity matrix

In summary, the Kronecker delta and the identity matrix may seem similar in definition and notation, but they are actually fundamentally different. The Kronecker delta is a function that returns 1 or 0 based on two indices, while the identity matrix is a matrix with elements that can be expressed using the Kronecker delta. However, the Kronecker delta itself is not a matrix, and cannot be simplified to the identity matrix. They serve different purposes, with the Kronecker delta being used for componentwise operations on matrices.
  • #1
loom91
404
0
Hi,

As in the title, what's the difference between the Kronecker delta and the identity matrix? They seem to have the exact same definition, so why are they differentiated? Thanks.

Molu
 
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  • #2
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! However, the elements of an identity matrix, which can be expressed with two indices as well, can be written with the kronecker delta since [tex]\left( {I_n } \right)_{i,j} = \delta _{i,j}[/tex].
 
  • #3
But the Kronecker delta considered as a whole is no more a single number than a matrix of numbers is a single number! The elements of an identity matrix are equivalent to the elements of the Kronecker delta, so why make the difference?
 
  • #4
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.
 
  • #5
So the only difference is in terminology? They even have the same notation!
 
  • #6
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. It is not a matrix. 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.
 
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  • #7
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 said), the kronecker delta is not.
 

What is the Kronecker delta?

The Kronecker delta, denoted by δ, is a mathematical symbol used to represent the discrete analogue of the Dirac delta function. It is defined as 1 if the two indices are equal, and 0 otherwise.

What is the identity matrix?

The identity matrix, denoted by I or sometimes by 1, is a square matrix with 1s on the main diagonal and 0s everywhere else. It is analogous to the number 1 in regular multiplication as it has the property that when multiplied by any other matrix, it remains unchanged.

What is the difference between Kronecker delta and identity matrix?

The Kronecker delta is a mathematical symbol used to represent a specific value (1 or 0) in a matrix, while the identity matrix is an actual matrix with specific properties. The Kronecker delta is a single value, while the identity matrix is a collection of values.

When is the Kronecker delta used?

The Kronecker delta is commonly used in mathematics and physics to represent discrete functions, particularly in the study of matrices and tensors. It is also used in probability theory and statistics to represent the probability of an event occurring.

How is the identity matrix related to the Kronecker delta?

The identity matrix is related to the Kronecker delta as it can be written in terms of the Kronecker delta symbol. Specifically, the identity matrix can be expressed as the sum of Kronecker delta symbols, with each delta representing a different row or column of the matrix.

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