- #1

FeDeX_LaTeX

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**Homework Statement**

For fixed m ≥ 1, let ##\epsilon(i,j)## denote the m x m matrix ##\epsilon(i,j)_{rs} = \delta_{ir}\delta_{js}##, where i,j may denote any integers in the range 1 ≤ i,j ≤ m.

(a) When m = 4, write out all ##\epsilon(i,j)## explicitly and label them correctly.

**The attempt at a solution**

My course hasn't started yet and all I know about the Kronecker delta/Levi-Civita epsilon notation is what little I've seen in videos.

They haven't stated the parameters of r and s, but I guess that doesn't matter since ##\delta_{i,j} = 1## if and only if i = j, and 0 otherwise?

All I can think of at the moment is this (can anyone tell me if this is correct?):

##\epsilon(1,1)_{rs} = \delta_{1r}\delta_{1s} = \delta_{11}\delta_{11}##

##\epsilon(1,2)_{rs} = \delta_{1r}\delta_{2s} = \delta_{11}\delta_{22}##

##\epsilon(1,3)_{rs} = \delta_{1r}\delta_{3s} = \delta_{11}\delta_{33}##

(...) etc. making 16 values in total. Is this the kind of thing they're looking for?

I still can't quite understand what I'm doing -- how does ##\delta_{11}\delta_{22}## represent an entry in a 4 x 4 matrix, for instance? Does that mean that both the entries in row 1 column 1 AND row 2 column 2 have value 1?

(I apologise for deleting the 'relevant equations' section -- I literally don't know which ones are relevant.)