Let $M$ be a $n$ by $n$ matrix over a field $F$.
$M$ is diagonizable, i.e. $M=P DP^{-1}$ for some invertible matrix $P$ and some diagonal matrix $D$, if and only if there exists an eigenbasis.
I wonder if $M$ can be similar to some special matrix if and only if there exists a generalized eigenbasis. Note a generalized eigenvector $v$ for an eigenvalue $\lambda$ with algebraic multiplicity $c$ is defined as a vector which satisfies $ (M - \lambda I)^c v = 0$.
$M$ admits a Jordan decomposition $M=P JP^{-1}$ for some invertible matrix $P$ and some Jordan canonical form $J$, if and only if the characteristic polynomial of $M$ can split into linear factors over $F$.
Since columns of $P$ form a generalized eigenbasis, "the characteristic polynomial of $M$ can split into linear factors over $F$" implies "there exists a generalized eigenbasis", but I wonder if the reverse is false, i.e. "there exists a generalized eigenbasis" doesn't necessarily imply "the characteristic polynomial of $M$ can split into linear factors over $F$"?
Thanks!