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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 a linear map.\nLet $\\lambda$ be an [[eigenvalue|0BQAA2BK3Q]] of $T$.\nThe *algebraic multiplicity* of $\\lambda$ is the largest integer $m \\ge 1$ such that $(t - \\lambda)^m$ divides the [[characteristic polynomial|0J5EMJRRFM]] $p_T(t)$.\n\n## Remarks\n\nThe sum of the algebraic multiplicities of all eigenvalues is at most $\\dim V$, with equality when $p_T$ splits over $F$.\nFor $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$, the eigenvalue $1$ has algebraic multiplicity $2$.\nThe algebraic multiplicity is at least the [[geometric multiplicity|099TWVJ909]].\n\n## Explanations\n\n### L1 human\n\nAn eigenvalue can be a root of the characteristic polynomial several times.\nThe algebraic multiplicity counts how many times.\nFor the identity matrix of size $n$, the eigenvalue $1$ has algebraic multiplicity $n$.\n",
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