next up previous contents
Next: Tchebycheff polynomials Up: Uniform strings Previous: Uniform strings   Contents

eigenvalues and eigenvectors

The equations for the wave matrices would all have the form

\begin{eqnarray*}
\left[ \begin{array}{ll}
x_{i+1} \\ x_i
\end{array} \righ...
...left[ \begin{array}{ll}
x_i \\ x_{i-1}
\end{array} \right],
\end{eqnarray*}



whose characteristic equation, or dispersion relation, could be written in one of the other of two forms, the first favoring trigonometry and the second, half-angles:

\begin{eqnarray*}
\mu^2 - \mu (\frac{m \lambda}{k} - 2) + 1 & = & 0, \\
(\surd\mu + \frac{1}{\surd\mu})^2 = \frac{m \lambda}{k}.
\end{eqnarray*}



Matrices of (unnormalized) row and column eigenvectors are as before,

\begin{eqnarray*}
{\rm (rows)} & = & \left[ \begin{array}{cc}
\mu_+ & -1 \\ \...
...begin{array}{cc}
\mu_+ & \mu_- \\ 1 & 1
\end{array} \right]
\end{eqnarray*}



making Sylverster's theorem read:

\begin{eqnarray*}
f(M) & = & \frac{f(\mu_+)}{(\mu_+^2 - 1)} \left[ \begin{array...
...ight] \left[ \begin{array}{cc}
\mu_- & -1 \end{array} \right]
\end{eqnarray*}





Pedro Hernandez 2004-02-28