Diagonalizing and finding the eigenvalues of matrix of type $T$.

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I have seen some of the solutions type within the math.stackexchange but didn't able to get the clear idea. Consider here n to be $\ge$ 5. $$ T = \begin{bmatrix} \alpha_1 & \beta & & & \beta\\ \beta & \alpha_2 & \beta & & \\ & \ddots & \ddots & \ddots & \\ & & \beta & \alpha_{n-1} & \beta \\\beta & & & \beta & \alpha_{n} \end{bmatrix} $$ References will also be helpful. But I tried the some which I didn't understand (already mentioned by you). Giving an example will be worth full.