---
tag: 667YZTHE96
kind: concept
name: Central polynomial
msc: [16R10, 16R20]
sources:
  - url: https://en.wikipedia.org/wiki/Central_polynomial
same_as:
  wikidata: Q18345072
  wikipedia: Central polynomial
---

## Summary

A central polynomial for the matrix ring $M_n(k)$ is a noncommutative polynomial whose values on $n \times n$ matrices are always scalar matrices, but not always zero.
Their existence for every $n$ was proved independently by Formanek and by Razmyslov.
Central polynomials produce many central elements in [[polynomial identity rings|C0EDT81EGH]], and they are a main tool in the structure theory of these rings.

## Explanations

### L1 human

The values of most expressions in matrices, such as $xy - yx$, are ordinary matrices that are not multiples of the identity.
A central polynomial is a special expression whose value is always a multiple of the identity matrix, whatever matrices one substitutes, and is sometimes nonzero.
For $2 \times 2$ matrices, $(xy - yx)^2$ is such an expression.
For example, with $x = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ one gets $xy - yx = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, whose square is the identity matrix.