Trace of product the of tensors

In summary, a tensor is a mathematical object used to represent relationships between quantities in a system. The trace of a tensor is the sum of its diagonal elements and can be calculated by summing the diagonal elements of its matrix representation. It provides information about the magnitude of the tensor and can simplify calculations involving tensors. The trace of a product of tensors can be zero in certain cases, such as when the tensors are anti-symmetric or arranged in a way that cancels out the diagonal elements.
  • #1
sarrfriend
6
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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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  • #2
welcome to pf!

hi sarrfriend! welcome to pf! :smile:

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

1. What is a tensor?

A tensor is a mathematical object that can be represented as a multidimensional array of numbers. It is used to describe the relationships between different quantities in a system and is a fundamental concept in fields such as physics, engineering, and computer science.

2. What is the trace of a tensor?

The trace of a tensor is the sum of its diagonal elements. In other words, it is the sum of the elements that lie on the main diagonal when the tensor is represented as a matrix. It is a scalar quantity that provides information about the overall magnitude of the tensor.

3. How is the trace of a tensor calculated?

The trace of a tensor can be calculated by summing the diagonal elements of the tensor's matrix representation. However, for higher dimensional tensors, the trace may need to be calculated using more advanced mathematical techniques such as index notation or Einstein notation.

4. What is the significance of the trace of a product of tensors?

The trace of a product of tensors can provide valuable information about the relationship between the tensors and how they interact with each other. It can also be used to simplify complex calculations and equations involving tensors.

5. Can the trace of a product of tensors be zero?

Yes, the trace of a product of tensors can be zero if the tensors involved have certain properties or if they are arranged in a specific way. For example, the trace of a product of two anti-symmetric tensors will always be zero. Additionally, if the tensors are arranged in a way that cancels out the diagonal elements, the trace will also be zero.

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