What is transpose multiplier and forward multiplier?

Question

For linear system X = A*s, we define the forward and transpose multiplies Af and At as follows:

Af = @(s) A*s;  
At = @(s) A'*s; 

I want to know what is forward and transpose multiplies ? And what is the functionality of forward and transpose multiplies?

Answer 1

These lines define two anonymous functions in terms of $A$.

Both functions take matrices as their inputs (presumably, the intent is to use them on column vectors). For a matrix x, the input Af(x) returns the same thing as A*x, and the input At(x) returns the same thing as A'*x.

For example, you can try the following commands in the command line to see how it all works:

A=[1 1 0; 0 1 1; 0 0 1];
Af = @(s) A*s;  
At = @(s) A'*s;
x = [1; 0; 0];
Af(x)
At(x)

The last two lines should produce the outputs

1
0
0

and

1
1
0