Corollary 3

— Let $U$ and $V$ be two square matrices of order $n$ with coefficients in a commutative field $K$. If there exists an invertible square matrix $P$ over some extension $K'$ of $K$ such that $V = P^{-1}UP$, then there exists an invertible square matrix $Q$ over $K$ such that $V = Q^{-1}UQ$.

Let $E$ be a finite dimensional vector space over a commutative field $K$, let $(e_i)_{1 \leq i \leq n}$ be a basis of $E$ and let $u$ be an endomorphism of $E$. Then by the exact sequence (1) of VII, p. 29, the $K[X]$-module $E$, associated to $u$ is isomorphic to the quotient of the free $K[X]$-module $E[X]$, with basis $(1 \otimes e_i)$, by the image of $E[X]$ under the $K[X]$-linear map $X - \bar{u}$. The similarity invariants $q_i(X)$ of $u$ (VII, p. 32, Def. 2) are thus the invariant factors of $X - \bar{u}$ (VII, p. 22). Thus Prop. 6 of VII, p. 22, implies: