Carleman Matrix

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 $f(x)$ is defined as:
:M = \frac\left frac (f(x))^j \right ~,
so as to satisfy the (Taylor series) equation:
:(f(x))^j = \sum_^ M x^k.

For instance, the computation of $f(x)$ by
:f(x) = \sum_^ M x^k. ~
simply amounts to the dot-product of row 1 of M with a column vector \left ,x,x^2,x^3,...\right\tau.

The entries of M /math> in the next row give the 2nd power of $f(x)$:
:f(x)^2 = \sum_^ M x^k ~,
and also, in order to have the zeroth power of $f(x)$ in M /math>, we adopt the row 0 containing zeros everywhere except the first position, such that
:f(x)^0 = 1 = \sum_^ M x^k = 1+ \sum_^ 0* x^k ~.

Thus, the dot product of M /math> with the column vector \left ,x,x^2,...\right\tau yields the column vector \left ,f(x),f(x)^2,...\right\tau

: M * \left 1,x,x^2,x^3,...\right\tau = \left 1,f(x),(f(x))^2,(f(x))^3,...\right\tau.

Generalization

A generalization of the Carleman matrix of a function can be defined around any point, such as:
:M = M_x - x_0 _x + x_0/math>
or M = M /math> where $g(x) = f(x + x_0) - x_0$. This allows the matrix power to be related as:
:(M )^n = M_x - x_0 nM_x + x_0/math>

General Series

:Another way to generalize it even further is think about a general series in the following way:
:Let $f(x) = \sum_n c_n (f) \cdot \psi_n(x)$ be a series approximation of $f(x)$, where $\_n$ is a basis of the space containing $f(x)$
:We can define G = c_n(\psi_m \circ f), therefore we have \psi_m \circ f = \sum_n c_n (\psi_m \circ f) \cdot \psi_n = \sum_n G \cdot \psi_n, now we can prove that G \circ f= G \cdot G /math>, if we assume that $\_n$ is also a basis for $g(x)$ and $g(f(x))$.
:Let $g(x)$ be such that \psi_l \circ g = \sum_m G \cdot \psi_m where G = c_m (\psi_l \circ g).
:Now \sum_n G \circ f \psi_n
= \psi_l \circ (g \circ f)
= (\psi_l \circ g) \circ f
= \sum_m G (\psi_m \circ f)
= \sum_m G \sum_n G \psi_n
= \sum_ G G \psi_n
= \sum_ (\sum_m G G ) \psi_n
:Comparing the first and the last term, and from $\_n$ being a base for $f(x)$, $g(x)$ and $g(f(x))$ it follows that G \circ f=\sum_m G G = G \cdot G /math>

Examples

If we set $\psi_n(x) = x^n$ we have the Carleman matrix

If $\_n$ is an orthonormal basis for a Hilbert Space with a defined inner product $\langle f,g \rangle$, we can set $\psi_n = e_n$ and $c_n (f)$ will be $$. If $e_n(x) = e^$ we have the analogous for Fourier Series, namely $c_n (f) = \cfrac \int_^ \displaystyle f(x) \cdot e^dx$

Properties

Carleman matrices satisfy the fundamental relationship
*M \circ g= M ~,
which makes the Carleman matrix ''M'' a (direct) representation of $f(x)$. Here the term $f \circ g$ denotes the composition of functions $f(g(x))$.

Other properties include:
*\,M ^n= M n, where $\,f^n$ is an iterated function and
*\,M ^= M , where $\,f^$ is the inverse function (if the Carleman matrix is invertible).

Examples

The Carleman matrix of a constant is:
:M = \left(\begin
1&0&0& \cdots \\
a&0&0& \cdots \\
a^2&0&0& \cdots \\
\vdots&\vdots&\vdots&\ddots
\end\right)

The Carleman matrix of the identity function is:
:M_x = \left(\begin
1&0&0& \cdots \\
0&1&0& \cdots \\
0&0&1& \cdots \\
\vdots&\vdots&\vdots&\ddots
\end\right)

The Carleman matrix of a constant addition is:
:M_x + x= \left(\begin
1&0&0& \cdots \\
a&1&0& \cdots \\
a^2&2a&1& \cdots \\
\vdots&\vdots&\vdots&\ddots
\end\right)

The Carleman matrix of the successor function is equivalent to the Binomial coefficient:
:M_x + x= \left(\begin
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\right)
:M_x + x = \binom

The Carleman matrix of the logarithm is related to the (signed) Stirling numbers of the first kind scaled by factorials:
:M_x log(1 + x)= \left(\begin
1&0&0&0&0& \cdots \\
0&1&-\frac&\frac&-\frac& \cdots \\
0&0&1&-1&\frac& \cdots \\
0&0&0&1&-\frac& \cdots \\
0&0&0&0&1& \cdots \\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end\right)
:M_x log(1 + x) = s(k, j) \frac

The Carleman matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials:
:M_x \log(1 - x)= \left(\begin
1&0&0&0&0& \cdots \\
0&1&\frac&\frac&\frac& \cdots \\
0&0&1&1&\frac& \cdots \\
0&0&0&1&\frac& \cdots \\
0&0&0&0&1& \cdots \\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end\right)
:M_x \log(1 - x) = , s(k, j), \frac

The Carleman matrix of the exponential function is related to the Stirling numbers of the second kind scaled by factorials:
:$M_x$exp(x) - 1= \left(\begin
1&0&0&0&0& \cdots \\
0&1&\frac&\frac&\frac& \cdots \\
0&0&1&1&\frac& \cdots \\
0&0&0&1&\frac& \cdots \\
0&0&0&0&1& \cdots \\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end\right)
:$M_x$exp(x) - 1 = S(k, j) \frac

The Carleman matrix of exponential functions is:
:$M_x$exp(a x)= \left(\begin
1&0&0&0& \cdots \\
1&a&\frac&\frac& \cdots \\
1&2a&2a^2&\frac& \cdots \\
1&3a&\frac&\frac& \cdots \\
\vdots&\vdots&\vdots&\vdots&\ddots
\end\right)
:$M_x$exp(a x) = \frac

The Carleman matrix of a constant multiple is:
:M_x x= \left(\begin
1&0&0& \cdots \\
0&c&0& \cdots \\
0&0&c^2& \cdots \\
\vdots&\vdots&\vdots&\ddots
\end\right)

The Carleman matrix of a linear function is:
:M_x + cx= \left(\begin
1&0&0& \cdots \\
a&c&0& \cdots \\
a^2&2ac&c^2& \cdots \\
\vdots&\vdots&\vdots&\ddots
\end\right)

The Carleman matrix of a function $f(x) = \sum_^f_k x^k$ is:
:M = \left(\begin
1&0&0& \cdots \\
0&f_1&f_2& \cdots \\
0&0&f_1^2& \cdots \\
\vdots&\vdots&\vdots&\ddots
\end\right)

The Carleman matrix of a function $f(x) = \sum_^f_k x^k$ is:
:M = \left(\begin
1&0&0& \cdots \\
f_0&f_1&f_2& \cdots \\
f_0^2&2f_0f_1&f_1^2+2f_0f_2& \cdots \\
\vdots&\vdots&\vdots&\ddots
\end\right)

Related matrices

The Bell matrix or the Jabotinsky matrix of a function $f(x)$ is defined as
:B = \frac\left frac (f(x))^k \right ~,
so as to satisfy the equation
:(f(x))^k = \sum_^ B x^j ~,
These matrices were developed in 1947 by Eri Jabotinsky to represent convolutions of polynomials. It is the transpose of the Carleman matrix and satisfy

B \circ g= B ~, which makes the Bell matrix ''B'' an ''anti-representation'' of $f(x)$.

See also

* Carlemann linearization
* Composition operator
* Function composition
* Schröder's equation
* Bell polynomials

Notes

References

* R Aldrovandi
Special Matrices of Mathematical Physics
Stochastic, Circulant and Bell Matrices, World Scientific, 2001.
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* 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.
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