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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 of $T$.\nThe *geometric multiplicity* of $\\lambda$ is $\\dim E_\\lambda$, the [[dimension|03GSZ8RTZJ]] of the [[eigenspace|0G0RFPG8HR]] $E_\\lambda = \\ker(T - \\lambda \\id_V)$.\n\n## Remarks\n\nIt is the maximal number of linearly independent eigenvectors for $\\lambda$.\nIt satisfies $1 \\le \\dim E_\\lambda \\le m$, where $m$ is the [[algebraic multiplicity|0FY42N02DV]] of $\\lambda$.\nFor $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$, the eigenvalue $1$ has geometric multiplicity $1$ and algebraic multiplicity $2$.\nA matrix is [[diagonalisable|07SQCNV233]] if and only if $p_A$ splits and the two multiplicities agree for each eigenvalue.\n\n## Explanations\n\n### L1 human\n\nThe geometric multiplicity counts the independent directions that the map stretches by $\\lambda$.\nIt can be smaller than the algebraic multiplicity.\nWhen it is smaller, the matrix cannot be diagonalised.\n",
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