Adjoints of products are reverse-ordered products of adjoints

Here we examine an example of the general idea that adjoints of products are reverse-ordered products of adjoints. For this example we use the Fourier transformation. No details of Fourier transformation are given here and we merely use it as an example of a square matrix $\bold F$.We denote the complex-conjugate transpose (or adjoint) matrix with a prime, i.e., $\bold F'$.The adjoint arises naturally whenever we consider energy. The statement that Fourier transforms conserve energy is $\bold y'\bold y=\bold x'\bold x$ where $\bold y= \bold F \bold x$.Substituting gives $\bold F'\, \bold F = \bold I$, which shows that the inverse matrix to Fourier transform happens to be the complex conjugate of the transpose of $\bold F$.

With Fourier transforms, zero padding and truncation are especially prevalent. Most subroutines transform a dataset of length of 2ⁿ, whereas dataset lengths are often of length $m \times 100$.The practical approach is therefore to pad given data with zeros. Padding followed by Fourier transformation $\bold F$can be expressed in matrix algebra as

$${\rm Program} \eq \bold F \ \left[ \begin{array} {c} \bold I \\ \bold 0 \end{array} \right]$$ (6)

According to matrix algebra, the transpose of a product, say $\bold A \bold B = \bold C$,is the product $\bold C' = \bold B' \bold A'$ in reverse order. So the adjoint subroutine is given by

$${\rm Program'} \eq \left[ \begin{array} {cc} \bold I & \bold 0 \end{array} \right] \ \bold F'$$ (7)

Thus the adjoint subroutine truncates the data after the inverse Fourier transform. This concrete example illustrates that common sense often represents the mathematical abstraction that adjoints of products are reverse-ordered products of adjoints. It is also nice to see a formal mathematical notation for a practical necessity. Making an approximation need not lead to collapse of all precise analysis.