In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone. Other uses of Carleman matrices occur in the theory of probability generating functions, and Markov chains.
Contents
Definition
The Carleman matrix of an infinitely differentiable function
so as to satisfy the (Taylor series) equation:
For instance, the computation of
simply amounts to the dot-product of row 1 of
The entries of
and also, in order to have the zero'th power of
Thus, the dot product of
Bell matrix
The Bell matrix of a function
so as to satisfy the equation
so it is the transpose of the above Carleman matrix.
Jabotinsky matrix
Eri Jabotinsky developed that concept of matrices 1947 for the purpose of representation of convolutions of polynomials. In an article "Analytic Iteration" (1963) he introduces the term "representation matrix", and generalized that concept to two-way-infinite matrices. In that article only functions of the type
Analytic Iteration Author(s): Eri Jabotinsky Source: Transactions of the American Mathematical Society, Vol. 108, No. 3 (Sep., 1963), pp. 457–477 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1993593 Accessed: 19/03/2009 15:57
Generalization
A generalization of the Carleman matrix of a function can be defined around any point, such as:
or
Matrix properties
These matrices satisfy the fundamental relationships:
which makes the Carleman matrix M a (direct) representation of
Other properties include:
Examples
The Carleman matrix of a constant is:
The Carleman matrix of the identity function is:
The Carleman matrix of a constant addition is:
The Carleman matrix of the successor function is equivalent to the Binomial coefficient:
The Carleman matrix of the logarithm is related to the (signed) Stirling numbers of the first kind scaled by factorials:
The Carleman matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials:
The Carleman matrix of the exponential function is related to the Stirling numbers of the second kind scaled by factorials:
The Carleman matrix of exponential functions is:
The Carleman matrix of a constant multiple is:
The Carleman matrix of a linear function is:
The Carleman matrix of a function
The Carleman matrix of a function
Carleman Approximation
Consider the following autonomous nonlinear system:
where
At the desired nominal point, the nonlinear functions in the above system can be approximated by Taylor expansion
where
Without loss of generality, we assume that
Applying Taylor approximation to the system, we obtain
where
Consequently, the following linear system for higher orders of the original states are obtained:
where
Employing Kronecker product operator, the approximated system is presented in the following form
where