Well-known and useful facts are:

- any symmetric matrix over $\mathbb R$ is semi-simple (i.e. diagonalizable), and
- any hermitean matrix over $\mathbb C$ is semi-simple.

I will loosely speak about the shape of a matrix and mean the existence of some (linear) relations between matrix-entries (or functions of the matrix-entries).

Question:Let $k$ be an algebraically closed field of characteristic $p$. Is there any result whatsoever, which says that a rich class of matrices of a given shape consists only of semi-simple matrices.

Since I am more interested in positive results, the notion of shape is kept flexible. However, if it could be proved that semi-simplicity is not implied by any shape in some reasonable class of shapes, this would be interesting as well.