# Trace of inverse is complex conjugate of trace

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This page describes a useful fact in character theory/linear representation theory arising from rudimentary linear algebra

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## Statement

Suppose is an invertible linear transformation of finite order acting on a finite-dimensional -vector space. Then, the trace of is the complex conjugate of the trace of .

## Facts used

## Proof

Using the above fact about eigenvalues, we see that 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 are, element-wise, the inverses of the eigenvalues of . Hence, the trace of is the sum of the inverses of the eigenvalues of , and hence equal to the sum of the complex conjugates of the eigenvalues of . Since complex conjugation is additive, the trace of is the complex conjugate of the trace of .