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

\begin{pmatrix} a_1 & b_{1} & 0 & ... & 0 \\\ b_{1} & a_2 & b_{2} & & ... \\\ 0 & b_{2} & a_3 & ... & 0 \\\ ... & & ... & & b_{n-1} \\\ 0 & ... & 0 & b_{n-1} & a_n \end{pmatrix}

We take:

\begin{pmatrix}b_1=b_2=b_3=....=b_{n-1}=constant=b\end{pmatrix}

and \begin{pmatrix}a_n=n \end{pmatrix}

Any help is appreciated.

is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal $n\times n$ matrix of the form :

\begin{pmatrix} 1 & b & 0 & ... & 0 \\\ b & 2 & b & & ... \\\ 0 & b & 3 & ... & 0 \\\ ... & & ... & & b \\\ 0 & ... & 0 & b & n \end{pmatrix}

Where $b$ is a constant.

Any help is appreciated.