The only methods I know for determining the minimal polynomial of a matrix involve first calculating its characteristic polynomial, then either testing all possible factors or looking at the generalised eigenspaces as you would for Jordan normal form.
Are there any other good algorithmic methods for calculating it? Ideally, I don't what anything involving row/column operations since I don't know my matrix explicitly, but can determine certain properties like its characteristic polynomial.
I would also be happy if you can give me any methods for calculating other properties of the minimal polynomial like its degree.
EDIT: For clarification, my matrix comes from a group action and is defined over a finite field, so I could also determine the order potentially, but this may be quite computationally intensive.