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  "body": "\n## Definition\n\nLet $F$ be a field, let $\\lambda \\in F$, and let $k \\ge 1$.\nThe *Jordan block* $J_k(\\lambda)$ is the $k \\times k$ matrix with entries\n$$(J_k(\\lambda))_{ij} = \\begin{cases} \\lambda \u0026 \\text{if } j = i, \\\\ 1 \u0026 \\text{if } j = i + 1, \\\\ 0 \u0026 \\text{otherwise}. \\end{cases}$$\n\n## Remarks\n\nFor example, $J_3(\\lambda) = \\begin{pmatrix} \\lambda \u0026 1 \u0026 0 \\\\ 0 \u0026 \\lambda \u0026 1 \\\\ 0 \u0026 0 \u0026 \\lambda \\end{pmatrix}$.\n$J_k(\\lambda) - \\lambda I_k$ is nilpotent of index $k$.\nIts only eigenvalue is $\\lambda$, with algebraic multiplicity $k$ and geometric multiplicity $1$.\nSome authors put the $1$ entries below the diagonal.\n\n## Explanations\n\n### L1 human\n\nA Jordan block is the simplest matrix that is not diagonalisable, when its size is at least $2$.\nIt has one eigenvalue and only one line of eigenvectors.\nEvery complex matrix is built from such blocks, by the Jordan normal form.\n",
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