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## 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 . We denote the complex-conjugate transpose (or adjoint) matrix with a prime, i.e., . The adjoint arises naturally whenever we consider energy. The statement that Fourier transforms conserve energy is where . Substituting gives , which shows that the inverse matrix to Fourier transform happens to be the complex conjugate of the transpose of .

With Fourier transforms, zero padding and truncation are especially prevalent. Most modules transform a dataset of length of ; whereas, dataset lengths are often of length . The practical approach is therefore to pad given data with zeros. Padding followed by Fourier transformation can be expressed in matrix algebra as (13)

According to matrix algebra, the transpose of a product, say , is the product in reverse order. Therefore, the adjoint routine is given by (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  Next: Nearest-neighbor coordinates Up: FAMILIAR OPERATORS Previous: Zero padding is the

2014-09-27