Basic operators and adjoints

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., $\bold F\T$. The adjoint arises naturally whenever we consider energy. The statement that Fourier transforms conserve energy is $\bold y\T\bold y=\bold x\T\bold x$ where $\bold y= \bold F \bold x$. Substituting gives $\bold F\T \bold F = \bold I$, 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 modules transform a dataset of length of $2^n$; 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

$${\rm Program} \eq \bold F \ \left[ \begin{array}{c} \bold I \\ \bold 0 \end{array} \right]$$ (13)

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

$${\rm Program}\T \eq \left[ \begin{array}{cc} \bold I & \bold 0 \end{array} \right] \ \bold F\T$$ (14)

Thus, the adjoint routine 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 the collapse of all precise analysis.

Basic operators and adjoints