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.
The Carleman matrix of an infinitely differentiable function is defined as:
so as to satisfy the (Taylor series) equation:
For instance, the computation of by
simply amounts to the dot-product of row 1 of with a column vector .
The entries of in the next row give the 2nd power of :
and also, in order to have the zero'th power of in , we adopt the row 0 containing zeros everywhere except the first position, such that
Thus, the dot product of with the column vector yields the column vector
The Bell matrix of a function is defined as
so as to satisfy the equation
so it is the transpose of the above Carleman matrix.
Eri Jabotinsky developed that concept of matrices 1947 for the purpose of representation of convolutions of polynomials. In 1963 he introduces the term "representation matrix", and generalized that concept to two-way-infinite matrices. In that article only functions of the type are discussed, but considered for positive *and* negative powers of the function. Several authors refer to the Bell matrices as "Jabotinsky matrix" since (D. Knuth 1992, W.D. Lang 2000) , and possibly this shall grow to a more canonical name.
A generalization of the Carleman matrix of a function can be defined around any point, such as:
or where . This allows the matrix power to be related as:
- Another way to generalize it even further is think about a general series in the following way:
- Let be a series approximation of , where is a basis of the space containing
- We can define , therefore we have , now we can prove that , if we assume that is also a basis for and .
- Let be such that where .
- Comparing the first and the last term, and from being a base for , and it follows that
If we set we have the Carleman matrix
If is an ortonormal basis for a Hilbert Space with a defined inner product , we can set and will be . If we have the analogous for Fourier Series, namely
These matrices satisfy the fundamental relationships:
which makes the Carleman matrix M a (direct) representation of , and the Bell matrix B an anti-representation of . Here the term denotes the composition of functions .
Other properties include:
- , where is an iterated function and
- , where is the inverse function (if the Carleman matrix is invertible).
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 is:
The Carleman matrix of a function is:
Consider the following autonomous nonlinear system:
where denotes the system state vector. Also, and 's are known analytic vector functions, and is the element of an unknown disturbance to the system.
At the desired nominal point, the nonlinear functions in the above system can be approximated by Taylor expansion
is the partial derivative of with respect to at and denotes the Kronecker product.
Without loss of generality, we assume that is at the origin.
Applying Taylor approximation to the system, we obtain
where and .
Consequently, the following linear system for higher orders of the original states are obtained:
, and similarly .
Employing Kronecker product operator, the approximated system is presented in the following form
, and and matrices are defined in (Hashemian and Armaou 2015).
- R Aldrovandi, Special Matrices of Mathematical Physics: Stochastic, Circulant and Bell Matrices, World Scientific, 2001. (preview)
- R. Aldrovandi, L. P. Freitas, Continuous Iteration of Dynamical Maps, online preprint, 1997.
- P. Gralewicz, K. Kowalski, Continuous time evolution from iterated maps and Carleman linearization, online preprint, 2000.
- K Kowalski and W-H Steeb, Nonlinear Dynamical Systems and Carleman Linearization, World Scientific, 1991. (preview)
- D. Knuth, Convolution Polynomials arXiv online print, 1992
- Jabotinsky, Eri: Representation of Functions by Matrices. Application to Faber Polynomials in: Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp. 546– 553 Stable jstor-URL