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!