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  "body": "\n## Definition\n\nLet $F$ be a field, let $n \\ge 1$, and let $p(t) = t^n + c_{n-1} t^{n-1} + \\cdots + c_1 t + c_0$ be a monic polynomial over $F$.\nThe *companion matrix* of $p$ is the $n \\times n$ matrix\n$$C(p) = \\begin{pmatrix} 0 \u0026 0 \u0026 \\cdots \u0026 0 \u0026 -c_0 \\\\ 1 \u0026 0 \u0026 \\cdots \u0026 0 \u0026 -c_1 \\\\ 0 \u0026 1 \u0026 \\cdots \u0026 0 \u0026 -c_2 \\\\ \\vdots \u0026 \u0026 \\ddots \u0026 \u0026 \\vdots \\\\ 0 \u0026 0 \u0026 \\cdots \u0026 1 \u0026 -c_{n-1} \\end{pmatrix}.$$\n\n## Remarks\n\nThe [[characteristic polynomial|0VSSPAFM2C]] and the [[minimal polynomial|03NDVCSHSB]] of $C(p)$ both equal $p$.\nIn the basis $e_1, \\dots, e_n$, the map of $C(p)$ sends $e_i$ to $e_{i+1}$ for $i \u003c n$.\nSome authors use the transpose of this matrix as the companion matrix.\nCompanion matrices are the blocks of the [[rational canonical form|042JWAR3Q0]].\n\n## Explanations\n\n### L1 human\n\nFor each monic polynomial, the companion matrix is a matrix with exactly that characteristic polynomial.\nIt shifts each basis vector to the next one, and the polynomial fixes the image of the last one.\nSo questions about roots of polynomials become questions about eigenvalues of matrices.\n",
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