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Eigenstructure

Starting with the defining equation for an eigenvector $ \underline{e}$ and its corresponding eigenvalue $ \lambda$ ,

$\displaystyle \mathbf{A}\underline{e}_i = {\lambda_i}\underline{e}_i,\quad i=1,2
$

we get, using Eq.$ \,$ (C.134),

$\displaystyle \left[\begin{array}{cc} gc & c-1 \\ [2pt] gc+g & c \end{array}\right] \left[\begin{array}{c} 1 \\ [2pt] \eta_i \end{array}\right] = \left[\begin{array}{c} {\lambda_i} \\ [2pt] {\lambda_i}\eta_i \end{array}\right]. \protect$ (C.137)

We normalized the first element of $ \underline{e}_i$ to 1 since $ g\underline{e}_i$ is an eigenvector whenever $ \underline{e}_i$ is. (If there is a missing solution because its first element happens to be zero, we can repeat the analysis normalizing the second element to 1 instead.)

Equation (C.138) gives us two equations in two unknowns:

$\displaystyle gc+\eta_i(c-1)$ $\displaystyle =$ $\displaystyle {\lambda_i}
\protect$ (C.138)
$\displaystyle g(1+c) +c\eta_i$ $\displaystyle =$ $\displaystyle {\lambda_i}\eta_i$ (C.139)

Substituting the first into the second to eliminate $ {\lambda_i}$ , we get

\begin{eqnarray*}
g+gc+c\eta_i &=& [gc+\eta_i(c-1)]\eta_i = gc\eta_i + \eta_i^2 (c-1)\\
\,\,\Rightarrow\,\,\eta_i &=& \frac{c(g-1)}{2(1-c)}
\pm \sqrt{\frac{c^2(1-g)^2 - 4g(1-c^2)}{(c-c)^2}}\\
&=& \frac{(g-1)c}{2(1-c)} \pm
j\sqrt{g\left(\frac{1+c}{1-c}\right)
- \frac{c^2(1-g)^2}{4(1-c)^2}}.
\end{eqnarray*}

As $ g$ approaches $ 1$ (no damping), we obtain

$\displaystyle \eta_i = \pm j\sqrt{\frac{1+c}{1-c}} \qquad \hbox{(when $g=1$)}.
$

Thus, we have found both eigenvectors:

\begin{eqnarray*}
\underline{e}_1&=&\left[\begin{array}{c} 1 \\ [2pt] \eta \end{array}\right],\qquad
\underline{e}_2=\left[\begin{array}{c} 1 \\ [2pt] -\eta \end{array}\right], \quad \hbox{where}\\
\eta&\isdef & \frac{(g-1)c}{2(1-c)}
+ j\sqrt{g\left(\frac{1+c}{1-c}\right)
- \frac{c^2(1-g)^2}{4(1-c)^2}}
\end{eqnarray*}

They are linearly independent provided $ \eta\neq0$ . In the undamped case ($ g=1$ ), this holds whenever $ c\neq -1$ . The eigenvectors are finite when $ c\neq 1$ . Thus, the nominal range for $ c$ is the interval $ c\in(-1,1)$ .

We can now use Eq.$ \,$ (C.139) to find the eigenvalues:

\begin{eqnarray*}
{\lambda_i}&=& gc+ \eta_i(c-1)\\
&=& gc+ \frac{(1-g)c}{2}\pm \sqrt{\left(\frac{c(1-g)}{2}\right)^2
- g (1-c^2)}\\
&=& c\left(\frac{1+g}{2}\right)
\pm j\sqrt{g(1-c^2) - \left[\frac{c(1-g)}{2}\right]^2}
\protect
\end{eqnarray*}



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2015-05-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA