How would you diagonalize the following infinite symmetric matrix?
$$T = \left[ \begin{array}{cccc} M \Omega^{2} + 2 \sigma & -c_{1} & -c_{2} & \cdots \\ -c_{1} & m_{1}w_{1}^{2} + 2 c_{1} & 0 & \cdots \\ -c_{2} & 0 & m_{2} w_{2}^{2} + 2 c_{2} & \\ \vdots & \vdots & & \ddots \\ \end{array} \right], $$ where $\sigma = \sum_{k} c_{k} < \infty$ and $c_{k}, w_{k}, m_{k}, M, \Omega $ are constants.