there is an unproved proposition in my lecture notes which states:
Let $A$ be an $n \times n$ matrix over $K$ that admits a Jordan basis. If $P$ is the matrix having a Jordan basis as columns, then $P^{-1}AP$ is the JCF of $A$.
First off I am not sure what is meant by $A$ admits a Jordan Basis. I assumed it means that the linear map that A represents admits a Jordan basis. But then I can't convince myself that this proposition is true. Please help!