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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, let $T \\colon V \\to V$ be an endomorphism, and let $\\lambda \\in F$.\nA *generalised eigenvector* of $T$ for $\\lambda$ is a vector $v \\in V$ with $v \\ne 0$ and\n$$(T - \\lambda \\id_V)^k (v) = 0$$\nfor some integer $k \\ge 1$.\nThe least such $k$ is the *rank* of the generalised eigenvector.\n\n## Remarks\n\nThe generalised eigenvectors of rank $1$ are the [[eigenvectors|0N6960W0BV]].\nIf a generalised eigenvector for $\\lambda$ exists, then $\\lambda$ is an eigenvalue.\nFor $A = \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$, the vector $e_2$ is a generalised eigenvector for $1$ of rank $2$, and it is not an eigenvector.\n\n## Explanations\n\n### L1 human\n\nSome matrices do not have enough eigenvectors to make a basis.\nGeneralised eigenvectors fill the gap: they reach zero after several applications of $T - \\lambda \\id$ instead of one.\nThey are the building blocks of the Jordan normal form.\n",
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