I wonder if there is an easy way to diagonalize the following matrix. One can make the following observation that the matrix can be partitioned into submatrices, so can you focus on only diagonalizing the $2 \times 2$ matrices? The matrix is given by $$A = \begin{pmatrix} (1-\dfrac{2m}{r}) & -\dfrac{2m}{r} & 0 & 0 \\ -\dfrac{2m}{r} & -(1+\dfrac{2m}{r}) & 0 & 0\\ 0 & 0 & -r^2 & 0\\ 0 & 0 & 0 & -r^2\sin^2(\theta) \end{pmatrix}.$$
Edit: Noticed I had a sign wrong in the second row second column.