DIFFERENTIAL EQUATIONS, CONVOLUTION, AND ADJOINTS

Simple one-dimensional equations of physics (such as vibrating strings) when discretized become tridiagonal matrices, each row with basically the same three elements (that arise from the expression of the second derivative operator). Where material properties are constant, the three elements are exactly the same from row to row. Likewise the convolution matrix representing filtering has rows identical to each other, except for the shift that aligns them along the diagonal. Examine a typical row of any such matrix.

$$\begin{eqnarray} \left[ \begin{array} {cccccc} a& b& c& & & \\ & a& b& c& & \... ...c& b& a& \\ & c& b&a \\ & & c&b \\ & & &c \end{array} \right]\end{eqnarray}$$ (1)

Notice that transposing such a matrix reverses the elements in the row. The reversal implies that the adjoint of convolution is crosscorrelation. Because of the reversal, the transpose of the first derivative matrix is the negative of the first derivative matrix itself (except at the edges). Generalizing, the discrete forms of differential equations representing physics in Cartesian coordinates show that any one-dimensional differential equation is its own adjoint if we set aside complications of boundaries, axis polarity, and material property variations. We will see something similar in higher dimensional spaces.