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alphaneutrino
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How can I show that trace is Invariant under the change of basis?
Trace is defined as the sum of the diagonal elements of a square matrix. In other words, it is the sum of the elements on the main diagonal of a matrix.
If the trace of a matrix remains the same when the basis for the vector space is changed, then it is said to be invariant under change of basis. This means that the trace is independent of the choice of basis for the vector space.
To show that trace is invariant under change of basis, you can use the fact that the trace of a matrix is equal to the sum of its eigenvalues. Since eigenvalues are independent of the basis, the trace will remain the same regardless of the basis chosen.
Proving that trace is invariant under change of basis is important because it is a fundamental property of matrices that allows for easier computation and manipulation. It also helps to simplify many calculations and proofs in linear algebra.
Yes, knowing that trace is invariant under change of basis allows you to use different bases that may be more convenient for calculation purposes. This can help simplify calculations involving trace and make them easier to understand and work with.