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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $n \\ge 1$.\nTwo $n \\times n$ matrices $A$ and $B$ over $R$ *commute* if\n$$AB = BA,$$\nwhere the products are [[matrix products|032Q9FB8MT]].\nA family of $n \\times n$ matrices is *commuting* if each two of its members commute.\n\n## Remarks\n\nExample: $A = \\begin{pmatrix} 1 \u0026 2 \\\\ 0 \u0026 1 \\end{pmatrix}$ and $B = \\begin{pmatrix} 3 \u0026 5 \\\\ 0 \u0026 3 \\end{pmatrix}$ over $\\R$ commute, since $AB = BA = \\begin{pmatrix} 3 \u0026 11 \\\\ 0 \u0026 3 \\end{pmatrix}$.\nNon-example: $A = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ and $B = \\begin{pmatrix} 0 \u0026 0 \\\\ 1 \u0026 0 \\end{pmatrix}$ do not commute, since $AB = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$ but $BA = \\begin{pmatrix} 0 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$.\nThe condition is often written $[A, B] = 0$, where $[A, B] = AB - BA$ is the commutator of $A$ and $B$.\nThe matrices that commute with a fixed matrix $A$ form the centraliser of $A$ in the matrix ring; it contains every polynomial in $A$.\nOver an algebraically closed field, a commuting family of matrices has a common [[eigenvector|0N6960W0BV]].\n",
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