# Characteristic polynomial of a tridiagonal matrix

Consider the polynomial recurrence

$$p_{k+1} (x) = (x - \alpha_{k+1})p_k(x) - \beta^2_{k+1}p_{k-1}(x), \quad (k=0,1,\ldots)$$

where $$p_0 = 1$$, $$p_{-1}=0$$, and $$\alpha_k$$ and $$\beta_k$$ are scalars.

Show that the roots of $$p_k(x)$$ are the eigenvalues of the below tridiagonal matrix

$$J_k = \begin{bmatrix} \alpha_1 & \beta_2 & & & \\ \beta_2 & \alpha_2 & \beta_3 & & & \\ & & \ddots & & \\ & & \beta_{k-1} & \alpha_{k-1} & \beta_k \\ & & & \beta_k & \alpha_k \end{bmatrix}$$

The recurrence relation can be obtained by the cofactor expansion of $J_{k+1}-xI_{k+1}$ along the last row (or column).
• Actually, the OP was almost right, the only correction is that $p_0=1$, which I've now edited to fix. Your version does not work. (This is an old question & answer, but it came up in search when I was looking just for this, so I'm commenting to prevent confusion for others in the future). – Meni Rosenfeld Aug 8 '20 at 20:35