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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]] $L = (l_{ij})$ of order $n$ over $R$ is *lower triangular* when $l_{ij} = 0$ for all $i \u003c j$.\nIt is *strictly lower triangular* when $l_{ij} = 0$ for all $i \\le j$.\n\n## Remarks\n\nA matrix is lower triangular if and only if its [[transpose|09DXWYRD43]] is [[upper triangular|0N4XY7Z3NN]].\nSums and products of lower triangular matrices of the same order are lower triangular.\nA system $Lx = b$ with $L$ lower triangular and invertible is solved by forward substitution.\nIn an [[LU decomposition|0RKG5YW66Y]], the factor $L$ is lower triangular.\n\n## Explanations\n\n### L1 human\n\nA lower triangular matrix has zeros everywhere above its diagonal.\nIts nonzero entries fill a triangle at the bottom left.\nFor example, $\\begin{pmatrix} 1 \u0026 0 \\\\ 2 \u0026 3 \\end{pmatrix}$ is lower triangular.\n",
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