Numerical Calculation of Eigenvalues of a large real Symmetric tridiagonal matrix If I have an $N \times N$ matrix where every entry is zero except for along the super-diagonal and sub-diagonal, where the each entry is the conjugate of the last, like the following $5 \times 5$ matrix
$$\begin{bmatrix}0 & (1-t) & 0 & 0 & 0 \\(1-t) & 0 & (1+t) & 0 & 0 \\0 & (1+t) & 0 & (1-t) & 0 \\0 & 0 & (1-t) & 0 & (1+t)\\0 & 0 & 0 & (1+t) & 0 \end{bmatrix}$$
where $t$ is some parameter. Imagine a much larger matrix following the same pattern.
Is there any way, numerical or otherwise to find the eigenvalues? 
 A: In general, the symmetric eigenvalue problem is solved by reducing the matrix to tridiagonal form and then applying the QR algorithm with implicit shifts. 
Only the last stage is relevant here and the run time is $O(n^2)$ as you only want the eigenvalues. I recommend starting with the routines implemented in LAPACK. A short list of relevant routines is given here
http://www.netlib.org/lapack/lug/node48.html
I see no way to exploit the very special substructure of your tridiagonal matrix in numerical computations.
A: This is not a full solution, but may give you some clues:
The trace of this matrix is zero, so the sum of eigenvalues should be zero. In fact for every odd power of the matrix the trace is zero.
If you multiply the matrix by itself the trace of new matrix is no longer zero, so the sum of eigenvalues of the squared matrix is not zero. In fact, again, every even power has non-zero trace, which leads me to believe that the eigenvalues will consist of non-zero pairs $\lambda, -\lambda$ and $0$ (maybe repeated values of $0$ for even order matrix, not sure).
So indeed, for the $5 \times 5$ matrix you have there, if I just use Mathematica to get the eigenvalues: $\left\{0,-\sqrt{t^2+3},\sqrt{t^2+3},-\sqrt{3 t^2+1},\sqrt{3 t^2+1}\right\}$. Maybe if you try with different orders of matrices you will be able to develop some formula in terms of $n$, where $n$ is the order of the matrix.
