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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 $\bold F$.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 modules 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
\begin{displaymath}
{\rm Program} \eq
\bold F \ 
 \left[ 
 \begin{array}
{c}
 \bold I \\  
 \bold 0
 \end{array} \right] \end{displaymath} (13)
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 routine is given by
\begin{displaymath}
{\rm Program'} \eq
 \left[ 
 \begin{array}
{cc}
 \bold I & \bold 0
 \end{array} \right] 
\
\bold F'\end{displaymath} (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 collapse of all precise analysis.


next up previous print clean
Next: Nearest-neighbor coordinates Up: FAMILIAR OPERATORS Previous: Zero padding is the
Stanford Exploration Project
4/27/2004