Commuting matrices

Summary

Two square matrices $A$ and $B$ commute when $AB = BA$. Matrix multiplication is not commutative in general, so commuting pairs are special. Commuting matrices behave much like numbers: for example, $e^{A + B} = e^{A} e^{B}$ for commuting matrices. Diagonalisable matrices can be diagonalised by one common basis exactly when they commute, by simultaneous diagonalisation.