On the characteristic roots of real matrices

Abstract:

If $A$ is a real $n \times n$-matrix whose absolute $|A|$ has spectral radius 1, and if $\varepsilon$ is a unimodular characteristic value of $A$, then all odd (respectively, even) powers of $\varepsilon$ are characteristic values of $A$ (respectively, of $|A|$). In particular, such $\varepsilon$ must be a $k$th root of unity for some $k, 1 \leqq k \leqq 2n$.