Trace of inverse is complex conjugate of trace

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Statement

Suppose $g$ is an invertible linear transformation of finite order acting on a finite-dimensional $\mathbb{C}$-vector space. Then, the trace of $g^{-1}$ is the complex conjugate of the trace of $g$.

Facts used

• Element of finite order is semisimple and eigenvalues are roots of unity

Proof

Using the above fact about eigenvalues, we see that $g$ is diagonalizable and all its complex eigenvalues are roots of unity. In particular, every eigenvalue is on the unit circle and its inverse equals its complex conjugate.

The eigenvalues of $g^{-1}$ are, element-wise, the inverses of the eigenvalues of $g$. Hence, the trace of $g^{-1}$ is the sum of the inverses of the eigenvalues of $g$, and hence equal to the sum of the complex conjugates of the eigenvalues of $g$. Since complex conjugation is additive, the trace of $g^{-1}$ is the complex conjugate of the trace of $g$.