Linear Algebra Question (Kronecker Delta?)

[FeDeX_LaTeX]

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.)

 

[tiny-tim]

Hi FeDeX_LaTeX!

$\epsilon(1,1)_{rs} = \delta_{1r}\delta_{1s} = \delta_{11}\delta_{11}$
$\epsilon(1,2)_{rs} = \delta_{1r}\delta_{2s} = \delta_{11}\delta_{22}$
$\epsilon(1,3)_{rs} = \delta_{1r}\delta_{3s} = \delta_{11}\delta_{33}$
(...) 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:

$\epsilon(1,1)_{11}\ \ \epsilon(1,1)_{12}$
$\epsilon(1,1)_{21}\ \ \epsilon(1,1)_{22}$

 

[FeDeX_LaTeX]

Hi FeDeX_LaTeX!


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

$\epsilon(1,1)_{11}\ \ \epsilon(1,1)_{12}$
$\epsilon(1,1)_{21}\ \ \epsilon(1,1)_{22}$

Thanks for the reply -- I'm confused... how did you know $\epsilon(1,1)$ should be a 2 x 2 matrix?

 

[Ray Vickson]

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.

 

[FeDeX_LaTeX]

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 $\epsilon(1,1)$?

 

[Office_Shredder]

$$\epsilon(1,1)$$ IS a 4x4 matrix. The r,s entry of it is given by $\delta_{1r} \delta_{s1}$.

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 $\delta_{1r} \delta_{s1}$ non-zero?

 

[Ray Vickson]

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.

 

[FeDeX_LaTeX]

$$\epsilon(1,1)$$ IS a 4x4 matrix. The r,s entry of it is given by $\delta_{1r} \delta_{s1}$.

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 $\delta_{1r} \delta_{s1}$ 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?

 

[Office_Shredder]

That's correct. The other ones can be solved similarly