Explicit Description of Centralizers for a Matrix

Tianhao Wang

paper

Explicit Description of Centralizers for a Matrix

arXiv:1910.13666

Abstract

Let $k$ be a field and $A \in M_{n} (k)$ be an $n \times n$ matrix. We denote $C_{M_{n} (k)} (A) = {B \in M_{n} (k) : B A = A B}$ 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.