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  "body": "\n## Definition\n\nLet $I = \\set{1, \\dots, n}$ be an index set.\nA *cluster tree* on $I$ is a rooted [[tree|062D9KXJB1]] whose nodes are nonempty subsets of $I$.\nIts root is $I$, and the children of each node partition that node.\nLet $\\mathcal{P}$ be a partition of $I \\times I$ into blocks $\\tau \\times \\sigma$ with $\\tau$ and $\\sigma$ nodes of the cluster tree.\nLet $\\mathcal{P} = \\mathcal{P}^{+} \\cup \\mathcal{P}^{-}$ be a split into *admissible* blocks $\\mathcal{P}^{+}$ and the remaining blocks $\\mathcal{P}^{-}$.\nLet $k \\ge 0$ be an integer.\nLet $M$ be a real [[matrix|0VD9ZSWWJG]] with rows and columns indexed by $I$.\nIt is a *hierarchical matrix* ($\\mathcal{H}$-matrix) of local rank $k$ for $\\mathcal{P}$ if each admissible block has small rank.\nThat is, $M|_{\\tau \\times \\sigma}$ has [[rank|0ZC9YTXZRY]] at most $k$ for every $\\tau \\times \\sigma \\in \\mathcal{P}^{+}$.\n\n## Remarks\n\nEach admissible block is stored as a product $X Y^T$ with $X$ of size $\\abs{\\tau} \\times k$ and $Y$ of size $\\abs{\\sigma} \\times k$.\nThe blocks in $\\mathcal{P}^{-}$ are stored in full; the partition is built so that they are small.\nIn applications each index $i$ carries a point $x_i \\in \\R^d$, and $\\Omega_\\tau$ is the set of points of a cluster $\\tau$.\nThe standard admissibility condition is $\\min(\\operatorname{diam} \\Omega_\\tau, \\operatorname{diam} \\Omega_\\sigma) \\le \\eta \\operatorname{dist}(\\Omega_\\tau, \\Omega_\\sigma)$ for a fixed $\\eta \u003e 0$.\nWith this condition, the matrices of asymptotically smooth kernels such as $1 / \\abs{x - y}$ are well approximated by $\\mathcal{H}$-matrices of small rank.\nExample: let $n = 2m$, split $I$ into halves $\\tau_1$ and $\\tau_2$, and call the two off-diagonal blocks admissible.\nA block diagonal matrix has zero off-diagonal blocks, of rank $0$, so it is an $\\mathcal{H}$-matrix of every local rank $k$.\nNon-example: the exchange matrix, with ones on the antidiagonal, has off-diagonal blocks of rank $m$.\nSo it is not an $\\mathcal{H}$-matrix of local rank $k \u003c m$ for this partition.\n",
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