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.
Definition
The Carleman matrix of a function
is defined as:
![{\displaystyle M[f]_{jk}={\frac {1}{k!}}\left[{\frac {d^{k}}{dx^{k}}}(f(x))^{j}\right]_{x=0}~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4403f8a3cf6059a61c85b4e2467c4a85f54b92e)
so as to satisfy the (Taylor series) equation:
![{\displaystyle (f(x))^{j}=\sum _{k=0}^{\infty }M[f]_{jk}x^{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03463b6cd637cee2f67b82f27d2090ea727f8911)
For instance, the computation of
by
![{\displaystyle f(x)=\sum _{k=0}^{\infty }M[f]_{1,k}x^{k}.~}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf258551c796e545bf7fb64d108e36a767bb4de)
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
:
![{\displaystyle f(x)^{2}=\sum _{k=0}^{\infty }M[f]_{2,k}x^{k}~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de8b11d0f8a91e4c2ced42170a60b0caf19a7384)
and also, in order to have the zero'th power of
in
, we aadopt the row 0 containing zeros everywhere except the first position, such that
![{\displaystyle f(x)^{0}=1=\sum _{k=0}^{\infty }M[f]_{0,k}x^{k}=1+\sum _{k=1}^{\infty }0*x^{k}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89492b7c2e86da2ea5950181e03c2401bba682e4)
Thus, the dot product of
with the column vector
yields the column vector
![{\displaystyle M[f]*\left[1,x,x^{2},x^{3},...\right]^{\tau }=\left[1,f(x),(f(x))^{2},(f(x))^{3},...\right]^{\tau }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11b855ee48ed632f75da2451b603eb0aa244af0f)
Bell matrix
The Bell matrix of a function
is defined as
![{\displaystyle B[f]_{jk}={\frac {1}{j!}}\left[{\frac {d^{j}}{dx^{j}}}(f(x))^{k}\right]_{x=0}~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6edf4d35ab6f9257f7c0341aa0aed08fcb35e32a)
so as to satisfy the equation
![{\displaystyle (f(x))^{k}=\sum _{j=0}^{\infty }B[f]_{jk}x^{j}~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8781733a538855e58051ba92c69fe22d63c9c1d0)
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. 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.
Generalization
A generalization of the Carleman matrix of a function can be defined around any point, such as:
![{\displaystyle M[f]_{x_{0}}=M_{x}[x-x_{0}]M[f]M_{x}[x+x_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60545c7d7eebc706af5c420424fc18ead0ffe7cc)
or
where
. This allows the matrix power to be related as:
![{\displaystyle (M[f]_{x_{0}})^{n}=M_{x}[x-x_{0}]M[f]^{n}M_{x}[x+x_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65827d744752dfa269cee519bb7d75f49f94575c)
Matrix properties
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:
Examples
The Carleman matrix of a constant is:
![{\displaystyle M[a]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&0&0&\cdots \\a^{2}&0&0&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27037f56eb81c02bca3637d7fa1a64a7acf69290)
The Carleman matrix of the identity function is:
![{\displaystyle M_{x}[x]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&1&0&\cdots \\0&0&1&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/060db1559fd634af4732397f145102b847ee28d0)
The Carleman matrix of a constant addition is:
![{\displaystyle M_{x}[a+x]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&1&0&\cdots \\a^{2}&2a&1&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3fea6f7f68d36e2bd565f790c580d6cf3638c7)
The Carleman matrix of a constant multiple is:
![{\displaystyle M_{x}[cx]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&c&0&\cdots \\0&0&c^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d29dc047a6112f5ff455b1dffd13a54b90102b18)
The Carleman matrix of a linear function is:
![{\displaystyle M_{x}[a+cx]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&c&0&\cdots \\a^{2}&2ac&c^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3518b703f7d0701200e12ef02d74528bb03450)
The Carleman matrix of a function
is:
![{\displaystyle M[f]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&f_{1}&f_{2}&\cdots \\0&0&f_{1}^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa0961a60884bc09a9405dea90f500a8747aea25)
The Carleman matrix of a function
is:
![{\displaystyle M[f]=\left({\begin{array}{cccc}1&0&0&\cdots \\f_{0}&f_{1}&f_{2}&\cdots \\f_{0}^{2}&2f_{0}f_{1}&f_{1}^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8287cbf2b6aa4ac5e2488a78832425dae8c6ae)
See also
References
- 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