Basic operators and adjoints

FAMILIAR OPERATORS

The simplest and most fundamental linear operators arise when a matrix operator reduces to a simple row or a column.

A row is a summation operation.

A column is an impulse response.

If the inner loop of a matrix multiply ranges within a

row, the operator is called sum or pull.

column, the operator is called spray or push.

Generally, inputs and outputs are high dimensional, such as signals or images. Push gives ugly outputs. Some output locations may be empty, each having an erratic number of contributions. Consequently, most data processing (adjoint) is done by pull.

A basic aspect of adjointness is that the adjoint of a row matrix operator is a column matrix operator. For example, the row operator $[a,b]$

$$y \eq \left[ a b \right] \left[ \begin{array}{l} x_1 \\ x_2 \end{array}\right] \eq a x_1 + b x_2$$ (1)

has an adjoint that is two assignments:

$$\left[ \begin{array}{l} \hat x_1 \\ \hat x_2 \end{arra... ... \left[ \begin{array}{l} a \\ b \end{array} \right] y$$ (2)

The adjoint of a sum of $N$ terms is a collection of $N$ assignments.

Basic operators and adjoints