---
tag: GJGEVMZ828
kind: concept
name: Generic matrix algebra
msc: [16R30, 16R10]
sources:
  - url: https://en.wikipedia.org/wiki/Generic_matrix_ring
same_as:
  wikidata: Q18207337
  wikipedia: Generic matrix ring
---

## Summary

The generic matrix algebra is the algebra generated by finitely many $n \times n$ matrices whose entries are independent commuting variables.
It is the "most general" algebra generated by $m$ matrices of size $n$.
For every commutative $k$-algebra $C$, the $k$-algebra generated by $m$ matrices in $M_n(C)$ is a quotient of it.
The quotient map substitutes elements of $C$ for the variables.
Over an infinite field $k$, it is the relatively free algebra of the variety of algebras that satisfy all identities of $M_n(k)$.
It is a domain.
It is a basic object in the theory of [[polynomial identity rings|C0EDT81EGH]] and in the invariant theory of matrices.

## Explanations

### L1 human

Fill a $2 \times 2$ matrix with four unknowns and a second one with four other unknowns.
The matrices you can build from these two by adding, multiplying and scaling form the generic matrix algebra.
Any rule that holds in this algebra holds for all pairs of $2 \times 2$ matrices, because you can substitute numbers for the unknowns.
Conversely, a rule that holds for all pairs of $2 \times 2$ matrices over an infinite field holds here, so this algebra records exactly the identities of $2 \times 2$ matrices.