Explicit Description of Centralizers for a Matrix

Let $k$ be a field and $A\in M_n(k)$ be an $n\times n$ matrix. We denote $C_{M_n(k)}(A) = \{B\in M_n(k) : BA = AB\}$ be its centralizers in $M_n(k)$. The dimension of the space of centralizer was already known by Frobenius. This paper will give the explicit $k$-basis for $C_{M_n(k)}(A)$ and also an algorithm (with polynomial complexity respect to multiplication in the field $k$) to construct the explicit basis. Lastly, the result can be used to solve a weaker version of the Wild Problem.