Exercise 2, § 5

a) Show that every square matrix over a field $K$ is similar to its transpose (reduce to the case of a Jordan matrix). b) Show that over the ring $\mathbf{Z}$ of rational integers, the matrix $\begin{pmatrix} 8 & 2 \\ 0 & 1 \end{pmatrix}$ is not similar to its transpose. c) Let $u$ be an endomorphism of a finite dimensional vector space over $K$. Then every eigenvalue $\lambda$ of $u$ is also an eigenvalue of the transpose endomorphism $'u$ of $E^*$. Moreover, if $A$ and $\mu$ are two distinct eigenvalues of $u$, if $x$ is an eigenvector of $u$ corresponding to $A$, and if $y^*$ is an eigenvector of $'u$ corresponding to $\mu$, then $\langle x, y^* \rangle = 0$.