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$4 \times 4$ wave matrices

A similar result applies to the $4 \times 4$ wave matrix of uniform second neighbor interactions,

\begin{eqnarray*}
\left[ \begin{array}{l} x_{i+2}\\ x_{i+1}\\ x_i\\ x_{i-1}\end...
...array}{l}x_{i+1}\\ x_i\\ x_{i-1}\\ x_{i-2} \end{array} \right].
\end{eqnarray*}



The sequence of indices is reversed by the self-inverse

\begin{eqnarray*}
R & = & \left[ \begin{array}{cccc}
0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0
\end{array} \right],
\end{eqnarray*}



so

\begin{eqnarray*}
\left[ \begin{array}{cccc}
0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \...
... 1 & 0 \\ 0 & 0 & 0 & 1 \\ d & c & b & a
\end{array} \right],
\end{eqnarray*}



while

\begin{eqnarray*}
\left[ \begin{array}{cccc}
a & b & c & d \\ 1 & 0 & 0 & 0 \...
...& 0 \\ 0 & 0 & 0 & 1 \\ 1 & -a & -b & -c
\end{array} \right].
\end{eqnarray*}



This time it is $d$ which should be $-1$, although $d = 1$ is also a possibility. Then it is also required that $a = c$, so the recursion relation

\begin{eqnarray*}
x_{i+2} + a x_{i+1} + b x_i + c x_{i-1} + d x_{i-2} = \lambda x_i
\end{eqnarray*}



has to be symmetric with respect to reversing the order of the indices.

The characteristic equation of a $4 \times 4$ wave matrix reads

\begin{eqnarray*}
\left\vert \begin{array}{cccc}
a - \mu & b & c & d \\ 1 & -...
... -\mu & 0 \\ 0 & 0 & 1 & -\mu
\end{array} \right\vert & = & 0
\end{eqnarray*}



Laplace expansion by the first row gives

\begin{eqnarray*}
(a - \mu) (-\mu)^3 - b (-\mu)^2 + c (-\mu) - d & = & 0
\end{eqnarray*}



Thus $d$ is the determinant, the product of all the roots, and should be $+1$. Putting the other symmetry condition, $a = c$,

\begin{eqnarray*}
(-\mu)^4 + a (-\mu)^3 - b(-\mu)^2 + a (-\mu) - d & = & 0 \\
...
... + \frac{1}{\mu})^2 - a (\mu + \frac{1}{\mu}) - (b + 2) & = & 0
\end{eqnarray*}



As before, the roots occur in reciprocal pairs, as they should when a matrix is equivalent to its inverse.


next up previous contents
Next: Dynamical matrix symmetry Up: Wave symmetry Previous: wave matrices   Contents
Pedro Hernandez 2004-02-28