Carleman matrix

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.

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:

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

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So for instance we have the computation of $f(x)$ by

$f(x) = \sum_{k=0}^{\infty} M[f]_{1,k} x^k.$

which is simply the dot-product of row 1 of $M[f]$ by a columnvector $\left[1,x,x^2,x^3,...\right]^\tau$

The entries of $M[f]$ of the next row give the 2nd power of $f(x)$:

$f(x)^2 = \sum_{k=0}^{\infty} M[f]_{2,k} x^k.$

and also, for to have the zero'th power of $f(x)$ in $M[f]$ we assume the row 0 containing zeros everywhere except the first position, such that

$f(x)^0 = 1 = \sum_{k=0}^{\infty} M[f]_{0,k} x^k = 1+ \sum_{k=1}^{\infty} 0* x^k$

Thus the dot-product of $M[f]$ with the column-vector $\left[1,x,x^2,...\right]^\tau$ gives the columnvector $\left[1,f(x),f(x)^2,...\right]^\tau$

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

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.

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:

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

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 $f(x)$. Here the term $f \circ g$ means the composition of functions $f(g(x))$

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).

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:

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

See also

• Bell polynomials

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)