Carleman matrix

From Wikipedia, the free encyclopedia

In mathematics, a Carleman matrix is a matrix that is used to convert function composition into matrix multiplication. They are used in iteration theory to find the continuous iteration of functions that cannot be iterated by pattern recognition alone. Other uses of Carleman matrices are in the theory of probability generating functions, and Markov chains.

1 Definition
2 Bell matrix
3 Generalization
4 Matrix properties
5 Categorical properties
6 Examples
7 See also
8 References

[edit] Definition

The Carleman matrix of a function $f (x)$ is defined as:

$M[f]_{jk} = \frac{1}{k!}\left[\frac{d^k}{dx^k} (f(x))^j \right]_{x=0}$

so as to satisfy the equation:

$(f(x))^j = \sum_{k=0}^{\infty} M[f]_{jk} x^k.$

[edit] Bell matrix

The Bell matrix of a function $f (x)$ is defined as:

$B[f]_{jk} = \frac{1}{j!}\left[\frac{d^j}{dx^j} (f(x))^k \right]_{x=0}$

so as to satisfy the equation:

$(f(x))^k = \sum_{j=0}^{\infty} B[f]_{jk} x^j$

which means it is basically the transpose of the Carleman matrix.

[edit] Generalization

A generalization of the Carleman matrix of a function can be defined around any point, such as:

$M[f]_{x_0} = M_x[x - x_0]M[f]M_x[x + x_0]$

or $M[f]_{x_0} = M[g]$ where $g (x) = f (x + x 0) - x 0$ . This allows the matrix power to be related as:

$(M[f]_{x_0})^n = M_x[x - x_0]M[f]^nM_x[x + x_0]$

[edit] Matrix properties

These matrices satisfy the fundamental relationships:

$M[f \circ g] = M[f]M[g]$
$B[f \circ g] = B[g]B[f]$

which makes the Carleman matrix M a (direct) representation of $f (x)$ , and the Bell matrix B an anti-representation of $f (x)$ .

Other properties include:

$\,M[f^n] = M[f]^n$ , where $\,f^n$ is function iteration and
$\,M[f^{-1}] = M[f]^{-1}$ , where $\,f^{-1}$ is the inverse function (if the Carleman matrix is invertible).

[edit] Categorical properties

The categorical approach to these properties is to let SetAn be the category of sets with analytic functions as morphisms, and let VecInf be the category of vector spaces with infinite matrices as morphisms between them. Using these categories, the Carleman matrix M is a (covariant) functor from SetAn to VecInf, and the Bell matrix B is a contravariant functor from SetAn to VecInf.

[edit] Examples

The Carleman matrix of a constant is:

$M[a] = \left(\begin{array}{cccc} 1&0&0& \cdots \\ a&0&0& \cdots \\ a^2&0&0& \cdots \\ \vdots&\vdots&\vdots&\ddots \end{array}\right)$

The Carleman matrix of the identity function is:

$M_x[x] = \left(\begin{array}{cccc} 1&0&0& \cdots \\ 0&1&0& \cdots \\ 0&0&1& \cdots \\ \vdots&\vdots&\vdots&\ddots \end{array}\right)$

The Carleman matrix of a constant addition is:

$M_x[a + x] = \left(\begin{array}{cccc} 1&0&0& \cdots \\ a&1&0& \cdots \\ a^2&2a&1& \cdots \\ \vdots&\vdots&\vdots&\ddots \end{array}\right)$

The Carleman matrix of a constant multiple is:

$M_x[cx] = \left(\begin{array}{cccc} 1&0&0& \cdots \\ 0&c&0& \cdots \\ 0&0&c^2& \cdots \\ \vdots&\vdots&\vdots&\ddots \end{array}\right)$

The Carleman matrix of a linear function is:

$M_x[a + cx] = \left(\begin{array}{cccc} 1&0&0& \cdots \\ a&c&0& \cdots \\ a^2&2ac&c^2& \cdots \\ \vdots&\vdots&\vdots&\ddots \end{array}\right)$

The Carleman matrix of a function $f(x) = \sum_{k=1}^{\infty}f_k x^k$ is:

$M[f] = \left(\begin{array}{cccc} 1&0&0& \cdots \\ 0&f_1&f_2& \cdots \\ 0&0&f_1^2& \cdots \\ \vdots&\vdots&\vdots&\ddots \end{array}\right)$

The Carleman matrix of a function $f(x) = \sum_{k=0}^{\infty}f_k x^k$ is:

$M[f] = \left(\begin{array}{cccc} 1&0&0& \cdots \\ f_0&f_1&f_2& \cdots \\ f_0^2&2f_0f_1&f_1^2& \cdots \\ \vdots&\vdots&\vdots&\ddots \end{array}\right)$