Difference between Kronecker delta and identity matrix

[loom91]

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

 

[TD]

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 $$\left( {I_n } \right)_{i,j} = \delta _{i,j}$$.

 

[loom91]

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?

 

[matt grime]

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.

 

[loom91]

So the only difference is in terminology? They even have the same notation!

 

[matt grime]

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.

 

[TD]

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.