Corollary 2

Let $r \in K(X)$ be a rational function over K. Then $u$ is substitutable (IV, p. 21) in $r$ if and only if each of its eigenvalues is. In this case the following formulae hold:

Write $r = p/q$ where $p$ and $q$ are coprime polynomials. Then $u$ is substitutable in $r$ if and only if $\det(q(u)) \neq 0$, so the first assertion follows from (8). By Cor. 1, we may suppose $q$ coprime to $\chi_u$, so by the Bezout identity there exist polynomials $g$ and $h$ such that $qg + h\chi_u = 1$. Then $q(\alpha_i)g(\alpha_i) = 1$ and $q(u)g(u) = 1$ by the Cayley-Hamilton theorem (III, p. 541). The stated formulae can then be obtained by applying formulae (6), (7) and (8) to $p(u)\ g(u) = r(u)$.