Is the trace of a matrix independent of basis?

• I
• Trixie Mattel
In summary, the trace of a matrix is independent of basis because it is equal to the sum of the eigenvalues of the operator it represents. This means that even if the matrix representation of an operator changes in different bases, the trace will remain the same.
Trixie Mattel
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

Trixie Mattel
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.

1. What is the trace of a matrix?

The trace of a matrix is the sum of the elements on the main diagonal, from the upper left corner to the lower right corner.

2. Why is the trace important?

The trace is important because it is a measure of the sum of the eigenvalues of a matrix, which provides useful information about the matrix's properties and behavior.

3. Is the trace of a matrix always a scalar value?

Yes, the trace of a matrix is always a scalar value, regardless of the size or dimensions of the matrix.

4. How is the trace affected by a change in basis?

The trace of a matrix is independent of basis, meaning it remains the same regardless of the basis used to represent the matrix. This is because the trace is a property of the matrix itself, not the basis used to represent it.

5. Can the trace of a matrix be negative?

Yes, the trace of a matrix can be negative. It is simply the sum of the elements on the main diagonal, so it can be positive, negative, or zero depending on the values in the matrix.

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