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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $m, n \\ge 1$.\nLet $A = (a_{ij})$ and $B = (b_{ij})$ be $m \\times n$ [[matrices|0VD9ZSWWJG]] over $R$.\nThe *Hadamard product* of $A$ and $B$ is the $m \\times n$ matrix $A \\circ B$ with entries\n$$(A \\circ B)_{ij} = a_{ij} b_{ij}$$\nfor $1 \\le i \\le m$ and $1 \\le j \\le n$.\n\n## Remarks\n\nExample: for $A = \\begin{pmatrix} 1 \u0026 0 \u0026 2 \\end{pmatrix}$ and $B = \\begin{pmatrix} 3 \u0026 5 \u0026 -1 \\end{pmatrix}$, the entries of $A \\circ B$ are $1 \\cdot 3$, $0 \\cdot 5$ and $2 \\cdot (-1)$, so $A \\circ B = \\begin{pmatrix} 3 \u0026 0 \u0026 -2 \\end{pmatrix}$.\nNon-example: a $2 \\times 2$ matrix and a $2 \\times 3$ matrix have no Hadamard product, because the two matrices must have the same size.\nThe Hadamard product is commutative, associative and distributive over addition.\nIts identity element is the $m \\times n$ matrix $J$ with all entries $1$, not the identity matrix.\nOther notations are $A \\odot B$ and $A * B$.\n",
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