Eigenvalues and eigenvectors of tridiagonal matrices

What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as $T = \begin{pmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-1} \\ & & & c_{n-1} & a_n \end{pmatrix}$.

If I set $a_i = 0$, do you know any previous results?

I know some results for simple cases like constant elements or symmetric matrices, but I would like to know if there are any results for more general cases.

• Possible duplicate of How can I calculate eigenvalues of a tridiagonal matrix? – Benjamin Steinberg Mar 2 '18 at 0:07
• I believe that the references mentioned in that post are related to symmetric tridiagonal matrices. – Guilherme Mar 2 '18 at 0:30
• I'll retract my vote. – Benjamin Steinberg Mar 2 '18 at 1:12
• I'm not sure it may be of any help, but if you further assume that $T$ is normal, which requires that $|c_i|=|b_i|$ for all $i$, then you can split $T$ as $T=S+A$ with $S$ (resp. $A$) tridiagonal symmetric (resp. antisymmetric) with $A,S$ diagonalizable in the same basis. This simplifies quite a lot the problem. In the non-normal case, the eigenvectors may not be orthogonal and I fear no particular structure would arise. – Adrien Hardy Mar 2 '18 at 1:30
• If $b_i$ and $c_i$ have the same sign for all $i$, then I think you can apply a diagonal rescaling to make it symmetric => it has real eigenvalues. – Federico Poloni Mar 2 '18 at 11:20