---
tag: 3KK7M27P8T
kind: concept
name: Carleman matrix
msc: [39B12, 13F25, 15B99]
sources:
  - url: https://en.wikipedia.org/wiki/Jabotinsky_matrix
---

## Summary

The Carleman matrix of a [[formal power series|033WGN00BF]] $f$ is the infinite matrix whose row $j$ lists the coefficients of the $j$-th power $f(x)^j$.
It turns composition of power series into [[matrix multiplication|032Q9FB8MT]].
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|AYVTR2ES5B]] of $f$ correspond to powers of a matrix.
This gives a linear-algebra approach to iteration and to [[fractional iteration|5M3THA1S6E]].

## Explanations

### L1 human

Composing power series by hand is messy, while multiplying matrices is routine.
The Carleman matrix records a power series together with all its powers in one table.
With these tables, substituting one series into another becomes a matrix product.
For $f(x) = ax$ the table is diagonal with entries $1, a, a^2, \ldots$, and composing $ax$ with $bx$ multiplies the diagonals.