# Why if $S$ is a linear operator on an odd-dimensional vector space, then it has a real eigenvalue?

as the title, why if $S$ is a linear operator on an odd-dimensional real vector space, then it has a real eigenvalue?

• Please, edit your post. – Sigur Dec 3 '12 at 18:26
• How do you compute the eigenvalue of $S$? Is it a root? – Sigur Dec 3 '12 at 18:27

Not true if your vector space is not real: for example, the map $t\mapsto it$ from $\mathbb C$ to $\mathbb C$ has the only eigenvalue $i$ and no other eigenvalue.