Prove that $\delta_{jl}\delta_{im}=\delta_{jm}\delta_{il}$

Question

Prove that $\delta_{jl}\delta_{im}=\delta_{jm}\delta_{il}$

In the video, he directly cancelled $$3\delta_{jl}\delta_{im}-3\delta_{jl}\delta_{im}$$

and similar terms. I was thinking if subscript between deltas are commutative.

In the video the equation was,

$$\delta_{il}\delta_{im}-\delta_{jm}\delta_{il}-\delta_{il}\delta_{jm}+3\delta_{jm}\delta_{il}+\delta_{im}\delta_{jl}-3\delta_{jl}\delta_{im}$$

here it's impossible to substract/cancel $3\delta_{jl}\delta_{im}$ and $3\delta_{jm}\delta_{il}$.

I am going to agree that he is wrong.

Answer 1

You might be asking the wrong question here...

The expression in the video is

$\underbrace{\delta_{jl}\delta_{im}+\delta_{im}\delta_{jl}-3\delta_{im}\delta_{jl}}_{\text{combine like terms}}~~~~\underbrace{-\delta_{il}\delta_{jm}-\delta_{jm}\delta_{il}+3\delta_{il}\delta_{jm}}_{\text{combine like terms}}$

Clearly, $\delta_{il}\delta_{jm}=\delta_{jm}\delta_{il}$ since this is just multiplying two numbers. So you can simplify the total expression, in the obvious way, to

$\delta_{il}\delta_{jm}-\delta_{jl}\delta_{im}$

The expression you ask in the title isn't actually right (as shown in a comment). The guy in the video also did not cancel the $3\delta$ terms, he just added/subtracted them with other like terms.

As for 'subscript of deltas are commutative', it is true that $\delta_{ij}=\delta_{ji}$, unless you mean something else?