Conditions for Schur decomposition and its generalization Let $M$ be a $n$ by $n$ matrix over a field $F$.
When $F$ is $\mathbb{C}$, $M$ always has a Schur decomposition, i.e. it is always unitarily similar to a triangular matrix, i.e. $M = U T U^H$ where $U$ is some unitary matrix and $T$ is a triangular matrix.


*

*I was wondering for an arbitrary field $F$, what are some conditions
for $M$ to admit Schur decomposition?

*Consider a generalization of Schur decomposition, $M = P T P^{-1}$
where $P$ is some invertible matrix and $T$ is a triangular matrix.
I was wondering what some conditions are for $M$ to admit such an
decomposition? 
Note that $M$ admit such an decomposition when $F$ is
$\mathbb{C}$, since it always has Schur decomposition.
Thanks!

If the characterisic polynomial factors in linear factors then the Jordan decomposition works as your triangular matrix.
If you have a similar triangular matrix then the characteristic polynomial of $M$ is the characteristic polynomial of $T$ which clearly factors into linear factors.
So, the criterion is exactly the same as for Jordan decomposition.
The similar triangular matrix is just a lazy variant of Jordan decomposition.
