Commuting matrices

Remarks

Example: $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 5 \\ 0 & 3 \end{pmatrix}$ over $\R$ commute, since $AB = BA = \begin{pmatrix} 3 & 11 \\ 0 & 3 \end{pmatrix}$. Non-example: $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ do not commute, since $AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ but $BA = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$. The condition is often written $[A, B] = 0$, where $[A, B] = AB - BA$ is the commutator of $A$ and $B$. The matrices that commute with a fixed matrix $A$ form the centraliser of $A$ in the matrix ring; it contains every polynomial in $A$. Over an algebraically closed field, a commuting family of matrices has a common eigenvector.