In my paper, I investigate the coordinates of the eigenvectors of a hollow tridiagonal hermitian matrix, which is defined as: \begin{align*} Q_n = \begin{pmatrix} 0 & q_{1,2} & 0 & 0 & 0\\ q_{2,1} & 0 & q_{2,3} & 0 & 0\\ 0 & q_{3,2} & \ddots & \ddots & \vdots\\ \vdots & \vdots & \ddots & \ddots & q_{n-1,n}\\ 0 & 0 & 0 & q_{n,n-1} & 0 \end{pmatrix} \end{align*}
where $q_{i, i + 1} = \overline{q_{i + 1, i}} \in \mathbb{C} \quad i=1 \ldots n$
The problem is to find such coefficients at which all coordinates of eigenvectors would be greater than precision $\varepsilon$.
But my current problem is custom calculation of eigenvectors coordinates with using the following relation: \begin{align}\label{eq:1} s_k=(-1)^{k-1} \cdot \alpha \cdot \dfrac{\Delta_{k-1}}{{Q_{1,2} \cdot Q_{2,3} \cdot \ldots \cdot Q_{k-1,k}}} \cdot e ^ {-j(\varphi_{1,2} + \varphi_{2,3} + \ldots + \varphi_{k-1,k})} \end{align}
where $k=2 \ldots n$, $\alpha - const$ - first coordinate of the eigenvector, $Q_{k-1, k} = | q_{k-1, k} |$, $\Delta_{k-1} = |Q_{k-1}-\lambda \cdot I| $
These relation comes from solving system of equations.
For Matrix(angle in degrees). I get matrix of Eigenvectors(transposed).
The calculations were done in Python using the Numpy library (I can attach the code if needed)
All checks say that I found the eigenvector matrix incorrectly.
I would also like to know the approach to solve the main problem (I was thinking of using Gershgorin circle theorem, to estimate eigenvalues and then try to solve the system of inequalities for coordinates).
