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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \\colon V \\to V$ be an endomorphism.\nThe *minimal polynomial* of $T$ is the monic polynomial $m_T \\in F[x]$ of least degree with $m_T(T) = 0$.\nIt is the monic generator of the ideal of [[annihilating polynomials|0PBN8VEKY8]] of $T$.\nThe minimal polynomial of a square matrix is defined in the same way.\n\n## Remarks\n\nThe minimal polynomial exists and is unique.\nIt [[divides each annihilating polynomial|0VTAD25A3G]], in particular the [[characteristic polynomial|0VSSPAFM2C]].\nFor $I_2$, the minimal polynomial is $x - 1$; for $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$, it is $(x - 1)^2$.\nSimilar matrices have the same minimal polynomial.\n\n## Explanations\n\n### L1 human\n\nThe minimal polynomial is the shortest polynomial relation that a matrix satisfies.\nAll other relations are multiples of it.\nIts roots are exactly the eigenvalues of the matrix.\n",
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