How can I diagonize this n×n matrix Hamiltonian?

[qubitor]

Homework Statement

sorry for my english..

I was asked to find a base to diagonize a Hamiltonian, which could been written in the given base as below:

n×n matrix

0 1 0 0 0 ... 0 0
1 0 1 0 0 ... 0 0
0 1 0 1 0 ... 0 0
0 0 1 0 1 ... 0 0
... ... ... ... ...
0 0 0 0 0 ... 0 1
0 0 0 0 0 ... 1 0

Homework Equations

In order to diagonize this Hamiltonian, I think one could calculate its eigenvalues in this base to get
eigenfuntions, hence one can use matrix of eigenfunctions to tranforms this base to obtain a new base which diagonize Hamiltonian as
λ1 0 ... 0
0 λ2 ... 0
...
0 ... λn

The Attempt at a Solution

I tried to calculate the determinant to obtain eignenvalues of Hamiltonian by
det|λId - H|=0
But it is too complicated and I didn't find a way to calculate it in n dimensions. Is there some way that I didn't know to calculate the determinant?

 

[qubitor]

is there someone could make it?

 

[vela]

You could try expressing H as $H_{ij} = \delta_{i (j-1)}+\delta_{i (j+1)}$.