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  "body": "\n## Definition\n\nLet $R$ be a ring and let $n \\ge 0$ be an integer.\nA [[square matrix|0JXZKX526R]] $D = (d_{ij})$ of order $n$ over $R$ is *diagonal* when $d_{ij} = 0$ for all $i \\ne j$.\nFor $d_1, \\dots, d_n \\in R$, the matrix $\\operatorname{diag}(d_1, \\dots, d_n)$ is the diagonal matrix with $d_{ii} = d_i$ for each $i$.\n\n## Remarks\n\nThe product of two diagonal matrices is diagonal: $\\operatorname{diag}(a_1, \\dots, a_n) \\operatorname{diag}(b_1, \\dots, b_n) = \\operatorname{diag}(a_1 b_1, \\dots, a_n b_n)$.\nOver a commutative ring, diagonal matrices commute with each other.\nThe [[determinant|00KDZRW4C0]] of a diagonal matrix is the product of its diagonal entries.\nA matrix that is similar to a diagonal matrix is [[diagonalisable|07SQCNV233]].\n\n## Explanations\n\n### L1 human\n\nA diagonal matrix has numbers only on its main diagonal, from the top left to the bottom right.\nAs a map, it stretches each coordinate axis by its own factor.\nDiagonal matrices are easy to add, multiply and raise to powers.\n",
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