Tridiagonal matrix with charcteristic polynomial is a polynomial $T_k(x)$ given recursively

2017-12-14 05:24:12

Let the polynomial sequence $T_k(x)$ as follow:

$T_0(x)=1$, $T_1(x)=x$ and $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$ for all $n\geq 1$.

Show the following:

1) For every $n\geq 2$ find a tridiagonal matrix $A_n$ so that its characteristic polynomial is $T_n(x)$.

2) Show that $A_n$ is similar to symmetric tridiagonal matrix.

The symmetric tridiagonal are the matrix in this link

I showed one recursive formula for the polynomial characteristic of a tridiagonal matrix but i not can using this for to show the points 1 and 2 before.

I wait that can you help me with hints or any thing...