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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\mathbb{K}$ be a field, and let $A$ be an $n \\times n$ [[matrix|0JXZKX526R]] over $\\mathbb{K}$.\nThe matrix $A$ is *reducible* if some $n \\times n$ [[permutation matrix|08SP3DF8NB]] $P$ gives\n$$P^T A P = \\begin{pmatrix} B \u0026 C \\\\ 0 \u0026 D \\end{pmatrix},$$\nwhere $B$ and $D$ are square matrices of size at least $1$.\nThe matrix $A$ is *irreducible* if it is not reducible.\n\n## Remarks\n\nLet $G(A)$ be the [[directed graph|0KFYGSHHF5]] with vertices $1, \\ldots, n$ and an edge from $i$ to $j$ whenever $a_{ij} \\ne 0$.\nThen $A$ is irreducible if and only if $G(A)$ is strongly connected.\nFor a real nonnegative matrix $A$, this is also equivalent to $(I + A)^{n-1}$ having all entries positive.\nExample: $\\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ is irreducible, because its graph has the edges $1 \\to 2$ and $2 \\to 1$.\nNon-example: $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ is reducible with $P = I$, since it is already block upper triangular with $1 \\times 1$ blocks.\nConventions for $1 \\times 1$ matrices differ between authors.\n",
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