Madagascar Programming Reference Manual |

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

where is the zero vector of length . One can readily verify that zero padding is a linear operation with operator matrix , where is the identity matrix and is the zero matrix, since
| |

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]

Madagascar Programming Reference Manual |

2011-07-02