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.

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