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 an infinitely differentiable 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 adopt 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. 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
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.
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: https://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:
![{\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)
General Series
- 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 ![{\displaystyle f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074)
- 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
.
- Now
![{\displaystyle \sum _{n}G[g\circ f]_{mn}\psi _{n}=\psi _{l}\circ (g\circ f)=(\psi _{l}\circ g)\circ f=\sum _{m}G[g]_{lm}(\psi _{m}\circ f)=\sum _{m}G[g]_{lm}\sum _{n}G[f]_{mn}\psi _{n}=\sum _{n,m}G[g]_{lm}G[f]_{mn}\psi _{n}=\sum _{n}(\sum _{m}G[g]_{lm}G[f]_{mn})\psi _{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b00c95ee186e5f7edd35a3d569a44b6116a7f0c)
- Comparing the first and the last term, and from
being a base for
,
and
it follows that ![{\displaystyle G[g\circ f]=\sum _{m}G[g]_{lm}G[f]_{mn}=G[g]\cdot G[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da354f5e5f9a1bba6f9fa933fa3a7249dd514e0d)
Examples
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
Matrix properties
These matrices satisfy the fundamental relationships:
![{\displaystyle M[f\circ g]=M[f]M[g]~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e00d6d5242cbe33b61cac226a4616334e0c3764)
![{\displaystyle B[f\circ g]=B[g]B[f]~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f01891f569a74b4504402346d43146891660285b)
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).
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 the successor function is equivalent to the Binomial coefficient:
![{\displaystyle M_{x}[1+x]=\left({\begin{array}{ccccc}1&0&0&0&\cdots \\1&1&0&0&\cdots \\1&2&1&0&\cdots \\1&3&3&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd6f916a9d60728ea980fae6cac9e59c61c578a)
![{\displaystyle M_{x}[1+x]_{jk}={\binom {j}{k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a24191e059d1693cc283f0e99fb9de0c8f473b0b)
The Carleman matrix of the logarithm is related to the (signed) Stirling numbers of the first kind scaled by factorials:
![{\displaystyle M_{x}[\log(1+x)]=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&-{\frac {1}{2}}&{\frac {1}{3}}&-{\frac {1}{4}}&\cdots \\0&0&1&-1&{\frac {11}{12}}&\cdots \\0&0&0&1&-{\frac {3}{2}}&\cdots \\0&0&0&0&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61170871a71104a2460fbca6ffedd2a2d18d37a5)
![{\displaystyle M_{x}[\log(1+x)]_{jk}=s(k,j){\frac {j!}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08aa68c6365aea61ce9dfca0b36f57ce265f7aed)
The Carleman matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials:
![{\displaystyle M_{x}[-\log(1-x)]=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&\cdots \\0&0&1&1&{\frac {11}{12}}&\cdots \\0&0&0&1&{\frac {3}{2}}&\cdots \\0&0&0&0&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46a8b609d493fb9b990b72713526cb0e214639db)
![{\displaystyle M_{x}[-\log(1-x)]_{jk}=|s(k,j)|{\frac {j!}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7dc6e5117d3f9472837983f4c5cf3fb0ac7f0d)
The Carleman matrix of the exponential function is related to the Stirling numbers of the second kind scaled by factorials:
![{\displaystyle M_{x}[\exp(x)-1]=\left({\begin{array}{cccccc}1&0&0&0&0&\cdots \\0&1&{\frac {1}{2}}&{\frac {1}{6}}&{\frac {1}{24}}&\cdots \\0&0&1&1&{\frac {7}{12}}&\cdots \\0&0&0&1&{\frac {3}{2}}&\cdots \\0&0&0&0&1&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60dcc5bc8118a4d12edab88fe4d1b55bdf45750b)
![{\displaystyle M_{x}[\exp(x)-1]_{jk}=S(k,j){\frac {j!}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88f9a96182d5979c945be545163e45b0ce248e6d)
The Carleman matrix of exponential functions is:
![{\displaystyle M_{x}[\exp(ax)]=\left({\begin{array}{ccccc}1&0&0&0&\cdots \\1&a&{\frac {a^{2}}{2}}&{\frac {a^{3}}{6}}&\cdots \\1&2a&2a^{2}&{\frac {4a^{3}}{3}}&\cdots \\1&3a&{\frac {9a^{2}}{2}}&{\frac {9a^{3}}{2}}&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba38bbc8a6ed091fce516b7d31e2fcf28d0cf561)
![{\displaystyle M_{x}[\exp(ax)]_{jk}={\frac {(ja)^{k}}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ee0d7ea2fd8ded162198b085659a006b84cbfc)
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}+2f_{0}f_{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f14d52bc552668f19232bd0c31b6d90279d1bfb5)
Carleman Approximation
Consider the following autonomous nonlinear system:
![{\displaystyle {\dot {x}}=f(x)+\sum _{j=1}^{m}g_{j}(x)d_{j}(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23fbf993b5945a96f6f3b801adfda65d1a8a635a)
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
![{\displaystyle f(x)\simeq f(x_{0})+\sum _{k=1}^{\eta }{\frac {1}{k!}}\partial f_{[k]}\mid _{x=x_{0}}(x-x_{0})^{[k]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e84b18169cca116cea6b7cab49bd085358d2edf6)
where
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
![{\displaystyle {\dot {x}}\simeq \sum _{k=0}^{\eta }A_{k}x^{[k]}+\sum _{j=1}^{m}\sum _{k=0}^{\eta }B_{jk}x^{[k]}dj}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24d25912c1223a9c1513f4235d2fdbcb42f77fed)
where
and
.
Consequently, the following linear system for higher orders of the original states are obtained:
![{\displaystyle {\frac {d(x^{[i]})}{dt}}\simeq \sum _{k=0}^{\eta -i+1}A_{i,k}x^{[k+i-1]}+\sum _{j=1}^{m}\sum _{k=0}^{\eta -i+1}B_{j,i,k}x^{[k+i-1]}d_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd0b8b3adffc2c4d6bacf9eb517a4471755af0a3)
where
, and similarly
.
Employing Kronecker product operator, the approximated system is presented in the following form
![{\displaystyle {\dot {x}}_{\otimes }\simeq Ax_{\otimes }+\sum _{j=1}^{m}[B_{j}x_{\otimes }d_{j}+B_{j0}d_{j}]+A_{r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65d5ed5157c6aaaa06fa46899e1a2f24fccf5fa6)
where
, and
and
matrices are defined in (Hashemian and Armaou 2015).[1]
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