---
tag: ARGNTXTAST
kind: definition
name: Generic matrix algebra
msc: [16R30, 16R10]
defines: GJGEVMZ828
uses: [04C5ZBG1WT, 0SDC934R52, 0JXZKX526R, 3CTW1P4HBP, 2P3Y2FRCSZ, C0EDT81EGH]
sources:
  - url: https://en.wikipedia.org/wiki/Generic_matrix_ring
---

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $m, n \ge 1$ be integers.
Let $P = k\big[x^{(l)}_{ij} : 1 \le l \le m, \ 1 \le i, j \le n\big]$ be the [[polynomial ring|0SDC934R52]] in $m n^2$ commuting variables.
For $1 \le l \le m$, the *generic matrix* $X_l \in M_n(P)$ is the [[matrix|0JXZKX526R]] whose $(i, j)$ entry is $x^{(l)}_{ij}$.
The *generic matrix algebra* $k\{X_1, \ldots, X_m\}$ is the $k$-subalgebra of $M_n(P)$ generated by $X_1, \ldots, X_m$ and the identity matrix.

## Remarks

The algebra $k\{X_1, \ldots, X_m\}$ is an [[associative algebra|3CTW1P4HBP]] over $k$, and it is a subring of $M_n(P)$, so it satisfies every [[polynomial identity|C0EDT81EGH]] of $M_n(P)$.
Let $\pi \colon k\langle x_1, \ldots, x_m \rangle \to k\{X_1, \ldots, X_m\}$ be the homomorphism from the [[free algebra|2P3Y2FRCSZ]] with $\pi(x_l) = X_l$.
If $k$ is infinite, the kernel of $\pi$ is the set of polynomial identities of $M_n(k)$ in $m$ variables.
Indeed, $f(X_1, \ldots, X_m) = 0$ means that each entry of the matrix $f(X_1, \ldots, X_m)$ is the zero polynomial.
Over an infinite field, a polynomial is zero exactly when it vanishes at all points, that is, when $f(a_1, \ldots, a_m) = 0$ for all $a_l \in M_n(k)$.

Example: for $n = 1$ the generic matrices are the variables $x^{(l)}_{11}$, and $k\{X_1, \ldots, X_m\}$ is the polynomial ring in $m$ variables.

Non-example: for $n \ge 2$, the scalar matrix $x^{(1)}_{11} I$ lies in $M_n(P)$ but not in $k\{X_1, \ldots, X_m\}$.
The entries of $f(X_1, \ldots, X_m)$ have degree-$1$ part equal to the corresponding entry of $\sum_l c_l X_l$, where $c_l$ is the coefficient of $x_l$ in $f$.
If $f(X_1, \ldots, X_m) = x^{(1)}_{11} I$, comparing the $(1, 1)$ entries gives $c_1 = 1$, and comparing the $(2, 2)$ entries gives $\sum_l c_l x^{(l)}_{22} = x^{(1)}_{11}$, which is impossible.
So the subalgebra generated by the $X_l$ over the commutative ring $P$ is strictly larger than the generic matrix algebra.

For $n \ge 2$ and $m \ge 2$, the generic matrix algebra is a noncommutative domain.