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  "body": "\n## Summary\n\nThe Carleman matrix of a [[formal power series|033WGN00BF]] $f$ is the infinite matrix whose row $j$ lists the coefficients of the $j$-th power $f(x)^j$.\nIt turns composition of power series into [[matrix multiplication|032Q9FB8MT]].\nIf $g$ has no constant term, the matrix of $f \\circ g$ is the product of the matrices of $f$ and $g$.\nSo, when $f$ has no constant term, the [[iterates|AYVTR2ES5B]] of $f$ correspond to powers of a matrix.\nThis gives a linear-algebra approach to iteration and to [[fractional iteration|5M3THA1S6E]].\n\n## Explanations\n\n### L1 human\n\nComposing power series by hand is messy, while multiplying matrices is routine.\nThe Carleman matrix records a power series together with all its powers in one table.\nWith these tables, substituting one series into another becomes a matrix product.\nFor $f(x) = ax$ the table is diagonal with entries $1, a, a^2, \\ldots$, and composing $ax$ with $bx$ multiplies the diagonals.\n",
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