Conditions for Schur decomposition and its generalization

Question

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.

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

2. 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!

Answer 1

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.

Answer 2

This is a thought. If you look at the construction of schur decomposition, at every step, one uses a new eigenvector to triangularize further and further (see here). So as long as the matrix has $n$ eigenvalues (distinct or repeated), I don't see any problem in extending schur decomposition to any field.