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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $\\lambda \\in F$.\nLet $T \\colon V \\to V$ be an endomorphism with [[minimal polynomial|03NDVCSHSB]] $m_T$.\nThen $\\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$ if and only if $m_T(\\lambda) = 0$.\n\n## Remarks\n\nIf $T(v) = \\lambda v$ with $v \\ne 0$, then $0 = m_T(T) v = m_T(\\lambda) v$, so $m_T(\\lambda) = 0$.\nConversely, if $m_T = (x - \\lambda) q$, then $q(T) \\ne 0$ and $T - \\lambda \\id_V$ kills the image of $q(T)$.\nSo the minimal and the characteristic polynomial have the same roots, possibly with different multiplicities.\n\n## Explanations\n\n### L1 human\n\nThe minimal polynomial and the characteristic polynomial have the same roots: the eigenvalues.\nOnly the multiplicities of the roots can differ.\nSo the minimal polynomial is a second way to find the eigenvalues.\n",
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