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  "body": "\n## Definition\n\nLet $m, n \\ge 1$ and let $A = (a_{ij})$ be a real $m \\times n$ [[matrix|0VD9ZSWWJG]].\nThe matrix $A$ is *nonnegative* if $a_{ij} \\ge 0$ for all $i$ and $j$.\nThe matrix $A$ is *positive* if $a_{ij} \u003e 0$ for all $i$ and $j$.\n\n## Remarks\n\nOne writes $A \\ge 0$ for a nonnegative matrix and $A \u003e 0$ for a positive matrix.\nMore generally, $A \\ge B$ means that $A - B \\ge 0$ entry by entry.\nExample: $\\begin{pmatrix} 1 \u0026 2 \\\\ 2 \u0026 1 \\end{pmatrix}$ is positive, hence nonnegative.\nIt is not [[positive semidefinite|01MMVW8CKC]], because it has the eigenvalue $-1$.\nNon-example: $\\begin{pmatrix} 1 \u0026 -1 \\\\ 0 \u0026 1 \\end{pmatrix}$ is not nonnegative, because its entry in row $1$ and column $2$ is $-1$.\nProducts and sums of nonnegative matrices are nonnegative.\n",
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