Hi, is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal matrix of the form :

$$ \mathcal{T}^{a}_n(p,q) = \begin{pmatrix} 0 & q & 0 & 0 &\cdots & 0 & 0 & 0 \\\ p & 0 & q & 0 &\cdots & 0 & 0 & 0 \\\ 0 & p & 0 & q &\cdots & 0 & 0 & 0 \\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots& \vdots \\\ 0 & 0 & 0 & 0 &\cdots & p & 0 & q \\\ 0 & 0 & 0 & 0 & \cdots & 0 & p & 0 \end{pmatrix} $$

where $p>0\ \ \& \ \ q > 0 $ and where there are $n$ rows and $n$ columns in the matrix above?

Furthermore is it possible to do the same for the equivalent matrix where periodic boundary conditions are implemented? i.e.

$$ \mathcal{T}^{b}_n(p,q) = \begin{pmatrix} 0 & q & 0 & 0 &\cdots & 0 & 0 & p \\\ p & 0 & q & 0 &\cdots & 0 & 0 & 0 \\\ 0 & p & 0 & q &\cdots & 0 & 0 & 0 \\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots& \vdots \\\ 0 & 0 & 0 & 0 &\cdots & p & 0 & q \\\ q & 0 & 0 & 0 & \cdots & 0 & p & 0 \end{pmatrix} $$

Thanks