What is a matrix algebra?
Solution
Let \(k\) be a commutative ring, usually a field. The set \[ M_n(k) \] of \(n\times n\) matrices with entries in \(k\) is an associative unital \(k\)-algebra:
- addition and scalar multiplication are entrywise;
- multiplication is matrix multiplication;
- the multiplicative identity is \(I_n\).
As a \(k\)-module it is free of rank \(n^2\), with basis given by the matrix units \[ E_{ij},\qquad 1\le i,j\le n, \] where \(E_{ij}\) has a single \(1\) in position \((i,j)\). Their multiplication is \[ E_{ij}E_{k\ell}=\delta_{jk}E_{i\ell}. \]
More generally, a matrix algebra over \(k\) can mean a \(k\)-subalgebra of some \(M_n(k)\), or specifically the full matrix algebra \(M_n(k)\) when the context is clear.