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Thread: how to show the eigenvalues of a jacobi matrix.

  1. #1
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    how to show the eigenvalues of a jacobi matrix.

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  2. #2
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    To find the eigenvalues, you need to evaluate $\displaystyle \Delta_n := \det(B-\lambda I)$. Expand along the top row to see that $\displaystyle \Delta_n = (a-\lambda)\Delta_{n-1} - bc\Delta_{n-2}$. Thus $\displaystyle \Delta_n$ satisfies the difference equation $\displaystyle \Delta_n - (a-\lambda)\Delta_{n-1} + bc\Delta_{n-2} = 0$, together with the initial conditions $\displaystyle \Delta_1 = a-\lambda$ and $\displaystyle \Delta_2 = (a-\lambda)^2 - bc$.

    The auxiliary equation $\displaystyle x^2 - (a-\lambda)x + bc = 0$ has solutions $\displaystyle x = \alpha\pm2ri\sqrt{1-(\alpha/r)^2}$, where $\displaystyle \alpha=\tfrac12(a-\lambda)$ and $\displaystyle r = \sqrt{bc}$. Standard techniques for solving difference equations (as described here, for example) show that $\displaystyle \Delta_n = \frac{r^n}{\sqrt{1-(\alpha/r)^2}}\sin\bigl((n+1)\theta\bigr)$, where $\displaystyle \cos\theta = \alpha/r$.

    The eigenvalues are given by $\displaystyle \Delta_n=0$. So they are the solutions of $\displaystyle \sin\bigl((n+1)\theta\bigr) = 0$, or $\displaystyle (n+1)\theta = k\pi$ $\displaystyle (1\leqslant k\leqslant n)$. Since $\displaystyle \cos\theta = \frac\alpha r = \frac{a-\lambda}{2r}$, it follows that $\displaystyle \lambda = a-2r\cos\theta = a - 2r\cos\bigl(\tfrac{k\pi}{n+1}\bigr)$. (You can then use the fact that $\displaystyle -\cos\phi = \cos(\pi-\phi)$ to write the eigenvalues as $\displaystyle \lambda_k = a + 2r\cos\bigl(\tfrac{k\pi}{n+1}\bigr)$.)

    Once you know the eigenvalues, it should be straightforward to get the eigenvectors, by solving the equations $\displaystyle (B-\lambda_k)u_k =0$.
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