If I were to conjugate by a matrix, what kind of matrix would it have to be in order to preserve trace? Does the trace of a matrix have anything to do with its spectrum? Any insights are appreciated, I'm just a bit confused.
2026-04-02 18:25:21.1775154321
Does conjugation preserve trace?
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For any square matrix $M$ whatsoever and any invertible matrix $Q$ of the same size, we have that the trace of $M$ equals the trace of $QMQ^{-1}$. No conditions on $Q$ or $M$. This follows immediately once you have the fact that $\text{tr}(AB) = \text{tr}(BA)$.
You asked for the relationship to the spectrum -- the trace is the sum of the eigenvalues (with appropriate multiplicities).