How to Find the Trace of ABA^-1 in Component Form?

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SUMMARY

The discussion focuses on finding the trace of the expression ABA-1 in component form, where A and B are matrices with components Aμν and Bμν. The indices are manipulated using the metric gμν and its inverse gμν. Participants explore the complexity of calculating the inverse of these generalized tensors and suggest utilizing the property tr(ABC) - tr(CAB) to simplify the problem. This approach aims to streamline the process of obtaining the trace without resorting to brute force methods.

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  • Knowledge of metric tensors and their inverses in the context of linear algebra.
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sarrfriend
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Let A, B be matrices with components Aμν , Bμν such that μ, ν = 0, 1, 2, 3. Indices are lowered and raised with the metric gμν and its inverse gμν. Find the trace of ABA-1 in component form?
Since A and B are generalized versions of tensors, finding their inverse becomes very tedious if we try to solve this by brute force, isn't it? Is there an easier way to find the solution?
 
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welcome to pf!

hi sarrfriend! welcome to pf! :smile:

hint: tr(ABC) - tr(CAB) = … ? :wink:​
 

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