What is the value of δ^{λ}_{μ} in 4 dimensions?

In summary, the value of ##\delta_\mu{}^\lambda## in the equation ##\eta_{\mu\nu} \eta^{\nu\lambda} = \delta_\mu{}^\lambda## is 1 if ##\mu = \lambda## and 0 otherwise. The equation represents the identity matrix in matrix notation and the double sum of the equation results in the trace of the 4x4 identity matrix, which is equal to 4.
  • #1
Dreak
52
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I'm starting to doubt about something:

In 4 dimension, what value has δ[itex]^{λ}_{μ}[/itex] in following equation:
ημ[itex]\nu[/itex]η[itex]\nu[/itex]λ = δ[itex]^{λ}_{μ}[/itex]

is it 4 or 1?

and IF it's 1, what is the difference that this equation:

ημσημσ

= 4?
 
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  • #2
##\delta_\mu{}^\lambda = 1## if ##\mu = \lambda## and 0 otherwise.

Only repeated indices are contracted, perhaps writing out the summations explicitly makes it clearer:

$$\eta_{\mu\nu} \eta^{\nu\lambda} \equiv \sum_{\nu = 1}^4 \eta_{\mu\nu} \eta^{\nu\lambda}$$
which is equal to 1 if ##\mu = \lambda## and 0 otherwise. In matrix notation, it basically says ##\eta^{-1} \eta = \mathbb{I}##.

If you also contract the two free indices, you get
$$\eta_{\mu\nu} \eta^{\nu\mu} \equiv \sum_{\mu = 1}^4 \sum_{\nu = 1}^4 \eta_{\mu\nu} \eta^{\nu\mu}$$
Replacing the inner sum by the delta, this is
$$\cdots = \sum_{\mu = 1}^4 \delta_\mu{}^\mu$$
and since the upper- and lower index are the same, the delta evaluates to 1 for every value of mu:
$$\cdots = \sum_{\mu = 1}^4 1 = 4$$

As $$\eta_{\mu\nu} \eta^{\nu\lambda}$$ is the ##(\mu, \lambda)## component of the matrix ##\eta^{-1} \eta##, in matrix notation the double sum this basically says
$$\sum_{i = 1}^4 (\eta^{-1} \eta)_{ii} = 4$$
which is indeed the trace of the 4x4 identity matrix.
 
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  • #3
Wow thanks, very clear explenation.
Thank you very much! :)
 

1. What is a contraction of matrices?

A contraction of matrices is a mathematical operation that involves multiplying two matrices and then taking the sum of the resulting matrix's diagonal elements.

2. How is a contraction of matrices different from multiplication?

A contraction is a specific type of matrix multiplication that only involves the diagonal elements of the resulting matrix, while traditional matrix multiplication involves all elements of the resulting matrix.

3. What is the purpose of a contraction of matrices?

The purpose of a contraction is to simplify and summarize the information in two matrices into a single matrix, making it easier to interpret and analyze the data.

4. Can a contraction of matrices be performed on matrices of any size?

Yes, a contraction can be performed on matrices of any size as long as they have the same dimensions.

5. Are there any practical applications of contractions of matrices?

Yes, contractions of matrices are commonly used in fields such as physics, engineering, and economics to analyze data and make predictions based on the simplified matrix representation.

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