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Mathematically speaking an operator is a function of a function, i.e. a rule (or mapping) according to which a function
is transformed into another function
. We use the notation
or simply
, where
denotes the operator. Examples of operators are the derivative, the integral, convolution (with a specific function), multiplication by a scalar and others. Note that in general the domains of
and
are not necessarily the same. For example, in the case of the derivative, the domain of
is the subset of the domain of
, in which
is smooth. In particular if
,
, then the domain of
is
.
An important class of operators are the linear operators. An operator
is linear if for any two functions
,
and any two scalars
,
,
. The derivative, integral, convolution and multiplication by scalar are all linear operators.
In the discrete world, operators act on vectors and linear operators are in fact matrices, with which the vectors are multiplied. (Multiplication by a matrix is a linear operation, since
). In fact many of the calculations performed routinely in science and engineering are essentially matrix multiplications in disguise. For example assume a vector
with length
(superscript
denotes transpose). Padding this vector with
zeros, produces another vector
with
Note that as in the case of functions, the domains of
and
are different:
(or more generally
), while
(or
).
Similarly, one can define convolution of
with
as the multiplication of
with
and many other operations as matrix multiplications. Other operators are the identity operator is the identity matrix
and is implemented by [sec:copy]copy.c and [sec:ccopy]ccopy.c and the null operator (or zero matrix
), which is implemented by [sec:adjnull]adjnull.c. For the rest of this introduction, the boldface notation will imply specifically discrete operators, while the normal fonts will imply operators on either continuous or discrete mathematical entities.
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| Madagascar Programming Reference Manual | |
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Next: Products of operators
Up: Introduction
Previous: Introduction
2011-07-02