Corollary 2

— Let u be an endomorphism of a finite dimensional vector space E over a field K, let $(q_1, \ldots, q_n)$ be the family of similarity invariants of u, let L be an extension of K, let $E_{(L)} = L \otimes_K E$ be the L-vector space induced from E by extension of scalars and let $u_{(L)} = 1_L \otimes u$ be the endomorphism of $E_{(L)}$ induced by u. Then the similarity invariants of $u_{(L)}$ are the images $\bar{q}_1, \ldots, \bar{q}_n$ of $q_1, \ldots, q_n$ in $L[X]$.

This follows immediately from Prop. 4 and the fact that the $L[X]$-modules $E_{(L)u_{(L)}}$ and $(K[X]/(q_i))_{(L)}$ are isomorphic to $L[X] \otimes_{K[X]} E_u$ and $L[X]/(\bar{q}_i)$ respectively.

Let U be a square matrix of order n with coefficients in a commutative field K. Then the similarity invariants of the endomorphism of $K^n$ defined by U are called the similarity invariants of $U$. It then follows from Cor. 1 above that two square matrices are similar if and only if they have the same similarity invariants, and that if $u$ is an endomorphism of a finite dimensional vector space $E$ over $K$, and $U$ is the matrix of $u$ with respect to some basis $B$ of $E$, then the similarity invariants of $U$ and $u$ coincide. By Cor. 1 and 2 above, we have: