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  "body": "\n## Summary\n\nA hierarchical matrix, or $\\mathcal{H}$-matrix, is a matrix split into a hierarchy of blocks.\nEach block that couples well-separated groups of indices has low [[rank|0ZC9YTXZRY]].\nSuch matrices are stored and multiplied with a vector in about $n \\log n$ operations instead of $n^2$.\nApproximate inverses and factorizations can be kept in the same format.\nThis gives fast solvers for the dense matrices of integral equations and for finite element matrices.\n\n## Explanations\n\n### L1 human\n\nThink of $n$ points in space and a matrix whose entry $(i, j)$ measures the interaction of point $i$ with point $j$, such as $1 / \\abs{x_i - x_j}$.\nTwo clusters of points that are far apart interact smoothly, so the block of the matrix linking them is close to a matrix of small rank.\nSuch a block can be stored with a few vectors instead of all its entries.\nOnly the blocks of nearby points are stored in full, and these are small.\nA hierarchical matrix applies this idea at all scales at once.\n",
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