I'm working through a problem in a textbook as follows:
"Consider the $d \times d$ Toeplitz matrix $$ A = \left[ \begin{array}{ccccc} 2 & 1 & 0 & \cdots & 0 \\ -1 &2 &1&\ddots&\vdots\\ 0&\ddots&\ddots&\ddots&0 \\ \vdots &\ddots&-1&2&1\\ 0&\cdots&0&-1&2 \end{array} \right].$$
Find explicitly all the eigenvalues of the Jacobi iteration matrix $B$. Conclude that this iterative scheme diverges."
Deriving the matrix $B$ as
$$ B = \left[ \begin{array}{ccccc} 0 & -\frac{1}{2} & 0 & \cdots & 0 \\ \frac{1}{2} &0 &-\frac{1}{2}&\ddots&\vdots\\ 0&\ddots&\ddots&\ddots&0 \\ \vdots &\ddots&\frac{1}{2}&0&-\frac{1}{2}\\ 0&\cdots&0&\frac{1}{2}&0 \end{array} \right]$$
and solving for the eigenvalues using the difference equation $q_{k+1} +2 \lambda q_k - q_{k-1} = 0$, I found the set of eigenvalues to be $$\lambda_{j} = i \cos(\frac{j\pi}{d+1}), \quad \quad j = 1,2, \dots, d $$
I then confirmed this fact using the eig
function in Matlab.
However, this leads to the conclusion that $\rho(B) = \max|\lambda_j| = |i\cos(\frac{\pi}{d+1})| < 1$, in which case the Jacobi Iteration should definitely converge (although maybe not very quickly). Why, then, does the textbook think that I should conclude the opposite?