What is an adjoint operator?

In mathematics the word ``adjoint'' has three meanings. One of them, the so-called Hilbert adjoint, is the one generally found in Physics and Engineering and it is the one used in this book.

In Linear Algebra is a different matrix, called the adjugate matrix. It is a matrix whose elements are signed cofactors (minor determinants). For invertible matrices, this matrix is the determinant times the inverse matrix. It is computable without ever using division, so potentially the adjugate can be useful in applications where an inverse matrix cannot. Unfortunately, the adjugate matrix is sometimes called the adjoint matrix particularly in the older literature. Because of the confusion of multiple meanings of the word adjoint, in the first printing of this book I avoided the use of the word, substituting the definition, ``conjugate transpose''. Unfortunately this was often abbreviated to ``conjugate'' which caused even more confusion.

EXERCISES:

1. Suppose a linear operator $\bold A$ has its input in the discrete domain and its output in the continuum. How does the operator resemble a matrix? Describe the operator $\bold A'$ which has its output in the discrete domain and its input in the continuum. To which do you apply the words ``scales and adds some functions,'' and to which do you apply the words ``does a bunch of integrals''? What are the integrands?
2. Examine the end effects in the programs contran() and convin(). Interpret differences in the adjoints.
3. An operator is ``self-adjoint'' if it equals its adjoint. Only square matrices can be self-adjoint. Prove by a numerical test that subroutine leaky() is self-adjoint.
4. Prove by a numerical test that the subroutine triangle() , which convolves with a triangle and then folds boundary values back inward, is self-adjoint.