Carleman matrix

Summary

The Carleman matrix of a formal power series $f$ is the infinite matrix whose row $j$ lists the coefficients of the $j$-th power $f(x)^j$. It turns composition of power series into matrix multiplication. If $g$ has no constant term, the matrix of $f \circ g$ is the product of the matrices of $f$ and $g$. So, when $f$ has no constant term, the iterates of $f$ correspond to powers of a matrix. This gives a linear-algebra approach to iteration and to fractional iteration.