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?
PART 1. We want to solve the equation (F): $p(X)=A$, in the unknown $X\in M_n(K)$, where $K$ is a field, $p\in K[x]$ and $A\in M_n(K)$. For the sake of simplicity, assume that $A$ is diagonalizable over $K$. Note that, if $X$ is a solution, then $XA=AX$. Then we may assume that $A=\lambda I$ and (F) is in the form $q(X)=0_n$.
PART 2. We want to solve the equation (E): $p(X)=0_n$ - where $p\in K[x]$ - in the unknown $X\in M_n(K)$.
i) $K$ is an algebraic closed field. The roots of $p$ are $(\alpha_i)_{i\leq s}$ with multiplicity $(r_i)_{i\leq s}$. Let $J_r$ be the nilpotent Jordan block of dimension $r$. Then $X$ is a solution of (E) IFF $X$ is similar to $diag(U_1,\cdots,U_k)$ for any choice of $k$ and of the dimension $n_j$ of $U_j$ satisfying $n_1+\cdots+n_k=n$ and for any choice of $U_j$ among the matrices of following forms: $\alpha_i I_{n_j}+J_{n_j}$ with $i\leq s$ and $n_j\leq r_i$.
ii) $K$ is a field not alg. closed. We factor $p$ in irreducible over $K$: $p=p_1^{r_1}\cdots p_s^{r_s}$. The result is similar to that of i). It suffices to choose $U_j$ among the companion matrices of ${p_i}^{q}$ where $q=n_j/degree(p_i)$ and $q\leq r_i$.
iii) $K$ is an euclidean ring (for example $\mathbb{Z}$). This case is much more difficult. We must seek a number of ideal classes ; using Magma software, we can do that but, morover, we must find one representant in each class, that is not obvious. For example, try to solve $A^3=I_n$ where $A\in M_n(\mathbb{Z})$.