The question is the following: given a matrix $$A=\begin{pmatrix} 1& 2 & & & & \\ 1& 0& 1 & & & \\ & 1& 0& 1 & &\\ & & \ddots & \ddots & \ddots & \\ & & & 1& 0 & 1\\ & & & & 1 &0 \end{pmatrix}.$$ Is it possible to give analytic expressions for the eigenvalues and eigenvectors of $A$?

Wang et. al [1] show that if the elements on the main diagonal are all 0, the eigenvalues and eigenvectors of $A$ can be expressed in trigonometric functions.

Thanks for your answer.

References

[1] W. Wang, C. M. Wang and S. L. Guo, On the walk matrix of the Dynkin graph $D_n$, Linear Algebra Appl. 653 (2022) 193-206.