I Is the trace of a matrix independent of basis?

AI Thread Summary
The trace of a matrix is indeed independent of the basis used, as it equals the sum of the eigenvalues of the corresponding operator. When an operator is represented by different matrices in different bases, the traces of these matrices remain equal. This means that while the matrix representation may change, the underlying operator does not. Therefore, the trace is a basis-independent property of the operator. Understanding this concept is essential in linear algebra and operator theory.
Trixie Mattel
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Hello,

Just wondering if the trace of a matrix is independent of basis, seeing as the trace of a matrix is equal to the sun of the eigenvalues of the operator that the matrix is a representation of.

Thank you
 
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mfb said:
I think you answered your own question, but a quick look at Wikipedia would lead to the same result.
Thank you mfb!
 
Trixie Mattel said:
Hello,

Just wondering if the trace of a matrix is independent of basis, seeing as the trace of a matrix is equal to the sun of the eigenvalues of the operator that the matrix is a representation of.

Thank you

What you might have meant is: if an operator is represented by a matrix ##M_1## in one basis, and ##M_2## in another basis, then is ##Tr(M_1) = Tr(M_2)##?

There is an important point that an operator does not change by a change of basis, but the matrix representing an operator may change from basis to basis.
 

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