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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.

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    $\begingroup$ Yes, you can diagonalize the upper 2x2-block (take a look here) $\endgroup$
    – StackTD
    Jan 12, 2017 at 10:12
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    $\begingroup$ Also, the eigenvalues of your top-left 2x2 block are $\pm \sqrt{1-4\frac mr +8\frac{m^2}{r^2}}$. $\endgroup$
    – Florian
    Jan 12, 2017 at 12:12

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