Explicit Description of Centralizers for a Matrix

TL;DR

This paper provides an explicit basis and an efficient algorithm for describing the centralizer of a matrix over a field, extending classical results and enabling solutions to related algebraic problems.

Contribution

It introduces a method to explicitly compute the basis of a matrix's centralizer and an algorithm with polynomial complexity for this task.

Findings

Explicit $k$-basis for the centralizer of a matrix

Polynomial-time algorithm for basis construction

Application to solving a weaker version of the Wild Problem

Abstract

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