Linear Algebra Question (Kronecker Delta?)

  • #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.)
 

Answers and Replies

  • #2
tiny-tim
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Hi FeDeX_LaTeX! :smile:
##\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?
No, they want you write out eg ##\epsilon(1,1)## as a matrix:

##\epsilon(1,1)_{11}\ \ \epsilon(1,1)_{12}##
##\epsilon(1,1)_{21}\ \ \epsilon(1,1)_{22}## :wink:
 
  • #3
FeDeX_LaTeX
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Hi FeDeX_LaTeX! :smile:


No, they want you write out eg ##\epsilon(1,1)## as a matrix:

##\epsilon(1,1)_{11}\ \ \epsilon(1,1)_{12}##
##\epsilon(1,1)_{21}\ \ \epsilon(1,1)_{22}## :wink:
Thanks for the reply -- I'm confused... how did you know ##\epsilon(1,1)## should be a 2 x 2 matrix?
 
  • #4
Ray Vickson
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Thanks for the reply -- I'm confused... how did you know ##\epsilon(1,1)## should be a 2 x 2 matrix?
He does not say that ##\epsilon(1,1)## is a 2 x 2 matrix; he just shows you some of what you need to compute. The question actually say m = 4, so you need a 4 x 4 matrix.
 
  • #5
FeDeX_LaTeX
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He does not say that ##\epsilon(1,1)## is a 2 x 2 matrix; he just shows you some of what you need to compute. The question actually say m = 4, so you need a 4 x 4 matrix.
So I'm completing a 4 x 4 matrix consisting of only ##\epsilon(1,1)##?
 
  • #6
Office_Shredder
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[tex]\epsilon(1,1)[/tex] IS a 4x4 matrix. The r,s entry of it is given by [itex] \delta_{1r} \delta_{s1} [/itex].

In particular you will notice that many of the entries of this matrix are zero. Can you identify what values of r and s make [itex] \delta_{1r} \delta_{s1} [/itex] non-zero?
 
  • #7
Ray Vickson
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So I'm completing a 4 x 4 matrix consisting of only ##\epsilon(1,1)##?
It reads to me like it wants you to write out all 16 of the 4 x 4 matrices, although I don't see what the point of doing that could possibly be---unless, maybe, there is a pattern that it is important to exploit later in some context.
 
  • #8
FeDeX_LaTeX
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[tex]\epsilon(1,1)[/tex] IS a 4x4 matrix. The r,s entry of it is given by [itex] \delta_{1r} \delta_{s1} [/itex].

In particular you will notice that many of the entries of this matrix are zero. Can you identify what values of r and s make [itex] \delta_{1r} \delta_{s1} [/itex] non-zero?
Sorry, I wasn't aware it represented a matrix -- I thought it just represented some number permutations. So ##\epsilon(1,1)## is just going to give me a 4 x 4 matrix with zero for every entry apart from the top-left, right?

r = s = 1 makes it non-zero?
 
  • #9
Office_Shredder
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That's correct. The other ones can be solved similarly
 

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