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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\sigma$ be a permutation of $\\{1, \\dots, n\\}$.\nThe *permutation matrix* of $\\sigma$ is the $n \\times n$ matrix $P_\\sigma$ with entries\n$$(P_\\sigma)_{ij} = \\begin{cases} 1 \u0026 \\text{if } i = \\sigma(j), \\\\ 0 \u0026 \\text{otherwise.} \\end{cases}$$\nA *permutation matrix* is a matrix of the form $P_\\sigma$.\n\n## Remarks\n\nThe matrix $P_\\sigma$ sends the standard basis vector $e_j$ to $e_{\\sigma(j)}$.\nWith this convention, $P_\\sigma P_\\tau = P_{\\sigma \\tau}$; some authors use the transposed convention.\nEach permutation matrix is invertible with $P_\\sigma^{-1} = P_\\sigma^T$, and $\\det P_\\sigma = \\sgn \\sigma$.\nThe matrix $\\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ is the permutation matrix that exchanges two coordinates.\n\n## Explanations\n\n### L1 human\n\nA permutation matrix is the identity matrix with its columns shuffled.\nMultiplying a vector by it shuffles the entries of the vector in the same way.\nEach row and each column has exactly one $1$.\n",
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