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In mathematics, a Carleman matrix is a matrix that is used to convert function composition into matrix multiplication. They are used in iteration theory to find the continuous iteration of functions that cannot be iterated by pattern recognition alone. Other uses of Carleman matrices are in the theory of probability generating functions, and Markov chains.In mathematics, function composition is the application of one function to the results of another. For instance, the functions f: X Y and g: Y Z can be composed by computing the output of g when it has an argument of f(x) instead of x. Intuitively, if z is a function g of y and y is a function f of x, then z is a function of x.