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The operation $ y_i = \sum_j b_{ij} x_j$ is the multiplication of a matrix $\bold B$ by a vector $\bold x$. The adjoint operation is $\tilde x_j = \sum_i b_{ij} y_i$. The operation adjoint to multiplication by a matrix is multiplication by the transposed matrix (unless the matrix has complex elements, in which case we need the complex-conjugated transpose). The following pseudocode does matrix multiplication $\bold y=\bold B\bold x$ and multiplication by the transpose $\tilde \bold x = \bold B' \bold y$:

		if operator itself 

then erase y
if adjoint
then erase x
do iy = 1, ny {
do ix = 1, nx {
if operator itself
y(iy) = y(iy) + b(iy,ix) $\times$ x(ix)
if adjoint
x(ix) = x(ix) + b(iy,ix) $\times$ y(iy)

Notice that the ``bottom line'' in the program is that $x$ and $y$ are simply interchanged. The above example is a prototype of many to follow, so observe carefully the similarities and differences between the operation and its adjoint.

A formal subroutine for matrix multiply and its adjoint is found below. The first step is a subroutine, adjnull(), for optionally erasing the output. With the option add=true, results accumulate like y=y+B*x.

void sf_adjnull (bool adj /* adjoint flag */, 
		 bool add /* addition flag */, 
		 int nx   /* size of x */, 
		 int ny   /* size of y */, 
		 float* x, 
		 float* y) 
/*< Zeros out the output (unless add is true). 
  Useful first step for any linear operator. >*/
    int i;
    if(add) return;
    if(adj) {
	for (i = 0; i < nx; i++) {
	    x[i] = 0.;
    } else {
	for (i = 0; i < ny; i++) {
	    y[i] = 0.;
The subroutine matmult() for matrix multiply and its adjoint exhibits the style that we will use repeatedly.
void matmult_lop (bool adj, bool add, 
		  int nx, int ny, float* x, float*y) 
/*< linear operator >*/
    int ix, iy;
    sf_adjnull (adj, add, nx, ny, x, y);
    for (ix = 0; ix < nx; ix++) {
	for (iy = 0; iy < ny; iy++) {
	    if (adj) x[ix] += B[iy][ix] * y[iy];
	    else     y[iy] += B[iy][ix] * x[ix];

Sometimes 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]
x_1 \\
a x_1 + b x_2
\end{displaymath} (1)

has an adjoint that is two assignments:
\hat x_1 \\
\hat x_2
a \\
\end{array} \right]
\end{displaymath} (2)

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

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