$2 \times 2$ wave matrices

Consider the $2 \times 2$ wave matrices

$$\begin{eqnarray*}\left[ \begin{array}{l} x_{i+1} \\ x_i \end{array} \right] & ... ...] \left[ \begin{array}{l} x_i \\ x_{i-1} \end{array} \right]. \end{eqnarray*}$$

If the direction of indexing is reversed, that can be described by a matrix

$$\begin{eqnarray*}\left[ \begin{array}{l} x_{i+1} \\ x_i \end{array} \right] & ... ...] \left[ \begin{array}{l} x_{i+1} \\ x_i \end{array} \right], \end{eqnarray*}$$

so the whole sequence could be reversed by writing

$$\begin{eqnarray*} \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right] \l... ...ht] \left[ \begin{array}{l} x_i \\ x_{i-1} \end{array} \right] \end{eqnarray*}$$

In the case $b = -1$, which was true in the wave matrix for a uniform chain, the same wave matrix serves for either direction. However, $b = -1$ makes the wave matrix unimodular, so that its eigenvalues occur in reciprocal pairs, and their logarithms, the wave numbers, are negatives of one another. In other words, similar waves propagate in opposite directions.