Levi-Civita symbol and Kronecker delta

AI Thread Summary
The discussion focuses on proving the relationship between the Levi-Civita symbol and the Kronecker delta, specifically the identity involving determinants. The user seeks guidance on how to demonstrate that the product of two Levi-Civita symbols equals the determinant of a matrix composed of Kronecker deltas. Another participant suggests expanding the determinant to find the connection. The conversation emphasizes the importance of understanding the properties of both symbols in proving the identity. Ultimately, the discussion highlights a common challenge in relating abstract algebraic concepts to concrete mathematical expressions.
typhoonss821
Messages
13
Reaction score
1
Hello everyone, I am stuck when I study Levi-Civita symbol.
The question is how to prove

\varepsilon_{ijk}\varepsilon_{lmn} = \det \begin{bmatrix}<br /> \delta_{il} &amp; \delta_{im}&amp; \delta_{in}\\<br /> \delta_{jl} &amp; \delta_{jm}&amp; \delta_{jn}\\<br /> \delta_{kl} &amp; \delta_{km}&amp; \delta_{kn}\\<br /> \end{bmatrix}

where \varepsilon_{ijk} represents Levi-Civita symbol and \delta_{il} represents kronecker symbol.

Thank you very much^^
 
Mathematics news on Phys.org
Have you already established the identity \epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}?
 
Yes I have, but I don't know how to relate it to determinant...
 
Well, you could just write out that determinant and see what happens.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Replies
3
Views
583
Replies
11
Views
3K
Replies
12
Views
4K
Replies
3
Views
2K
Replies
17
Views
2K
Replies
1
Views
14K
Replies
6
Views
8K
Replies
1
Views
2K
Replies
1
Views
29K
Back
Top