Carleman matrix - AbsoluteAstronomy.com

In mathematics, a Carleman matrix is a matrix that is used to convert function composition

Function composition

In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

into matrix multiplication

Matrix multiplication

In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...

. They are used in iteration theory to find the continuous iteration of functions that cannot be iterated by pattern recognition

Pattern recognition

In machine learning, pattern recognition is the assignment of some sort of output value to a given input value , according to some specific algorithm. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes...

alone. Other uses of Carleman matrices are in the theory of probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

generating functions, and Markov chains.

Definition

The Carleman matrix of a function

is defined as:

so as to satisfy the equation:

----

So for instance we have the computation of

which is simply the dot-product of row 1 of

by a columnvector

The entries of

of the next row give the 2nd power of

and also, for to have the zero'th power of

we assume the row 0 containing zeros everywhere except the first position, such that

Thus the dot-product of

with the column-vector

gives the columnvector

Bell matrix

The Bell matrix of a function

is defined as:

so as to satisfy the equation:

which means it is basically the transpose

Transpose

In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

of the Carleman matrix.

Generalization

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

where

. This allows the matrix power to be related as:

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

means the composition of functions

Other properties include:

, where is function iteration and
, where is the inverse function
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

(if the Carleman matrix is invertible).