It's me again. This time I got an idea from this question regarding polynomial equation systems arising from a matrix equation. Having read sporadically about polynomial equation systems and learned that the field to solve them is extremely theoretically challenging (algebraic geometry), I could not help but think about if one could go the other way around.
To try and design matrices in such a way that a monomial equation of the matrix would be the same as a polynomial equation system in the elements' field and then use a switch to some canonical basis ( ideally a diagonalization, but probably a block diagonalization could help quite a bit ). This would be useful for me as I have done much matrices and linear algebra.
- Would this be useful or is it only boring polynomial equation systems we would be able to express?
- If this is investigated, do you know any place to read more about how to do this?