We define an adjoint when we write a program that computes one. In an abstract logical mathematical sense, however, every adjoint is defined by a dot product test. This abstract definition gives us no clues how to code our program. After we have finished coding, however, this abstract definition (which is actually a test) has considerable value to us.
Conceptually, the idea of matrix transposition is simply aij'=aji. In practice, however, we often encounter matrices far too large to fit in the memory of any computer. Sometimes it is also not obvious how to formulate the process at hand as a matrix multiplication. (Examples are differential equations and fast Fourier transforms.) What we find in practice is that an application and its adjoint amounts to two subroutines. The first subroutine amounts to the matrix multiplication .The adjoint subroutine computes ,where is the conjugate-transpose matrix. Most methods of solving inverse problems will fail if the programmer provides an inconsistent pair of subroutines for and .The dot product test described next is a simple test for verifying that the two subroutines really are adjoint to each other.
The matrix expression may be written with parentheses as either or .Mathematicians call this the ``associative'' property. If you write matrix multiplication using summation symbols, you will notice that putting parentheses around matrices simply amounts to reordering the sequence of computations. But we soon get a very useful result. Programs for some linear operators are far from obvious, for example causint() . Now we build a useful test for it.
Of course passing the dot product test does not prove that a computer code is correct, but if the test fails we know the code is incorrect. More information about adjoint operators, and much more information about inverse operators is found in my other books, Earth Soundings Analysis: Processing versus inversion (PVI) and Geophysical Estimation by Example (GEE).